{
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  "category": "Set theory",
  "claimBoundary": "The selected declaration concludes that a bijective function α → β exists from hypotheses that particular functions f and g are injective. It does not assert that either given injection is itself bijective, and the page does not claim a new construction.",
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  "exactFormalStatement": "theorem Function.Embedding.schroeder_bernstein {α : Type u} {β : Type v} {f : α → β} {g : β → α} (hf : Function.Injective f) (hg : Function.Injective g) : ∃ h : α → β, Function.Bijective h",
  "family": "Cardinality and bijections",
  "id": "library.mathlib.schroeder-bernstein.v001",
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      "bijections",
      "cardinality",
      "infinite sets"
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    "plainLanguage": "If there is a one-to-one function from α into β and another one-to-one function from β into α, then there is a function from α to β that is both one-to-one and onto.",
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      "alt": "An ivory theorem poster shows injective maps from alpha to beta and beta to alpha, subordinate alternating-chain intuition, and a newly constructed bijective map pairing every displayed point exactly once.",
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      "caption": "Two particular injections in opposite directions imply the existence of a new bijection; neither input map is presented as bijective.",
      "description": "The top half displays separate forward and backward injections with unused target points. Alternating chains appear in a middle proof-intuition strip. The lower half reorganizes the two point fields into one exact pairing h, next to separate injective and surjective cues and an explicit scope footer.",
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      "sha256": "sha256:fda59680e23b08329183a0554951d7a0273a0212db6d417b3e98281dbe5d4daf",
      "title": "Schröder–Bernstein Theorem at a glance",
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      "transcript": "SET THEORY · CARDINALITY\nSCHRÖDER–BERNSTEIN THEOREM\nf : α → β · INJECTIVE\ng : β → α · INJECTIVE\n⇒ ∃ h : α → β\nh IS BIJECTIVE\nTWO ONE-TO-ONE MAPS\nOPPOSITE DIRECTIONS\nALTERNATING CHAINS · PROOF INTUITION\nONE NEW EXACT PAIRING\nINJECTIVE + SURJECTIVE\nEXACT SCOPE\nSome bijection h exists. Neither given injection is claimed bijective."
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    "visual": {
      "alt": "Opposing one-to-one arrows between emerald and cobalt nodes are reorganized into alternating chains and loops, then into a complete row-by-row pairing.",
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      "caption": "Two injections are decomposed into alternating components and resolved into a bijection.",
      "description": "Two source-and-target diagrams on the left show total injections in opposite directions. Alternating colored chains and a loop occupy the center. On the right, isolated emerald and cobalt nodes are paired exactly once by clean horizontal segments, with no within-column links.",
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      "title": "Schröder–Bernstein injection-to-bijection schematic",
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    "walkthrough": {
      "introduction": "Mathlib's selected theorem specializes a stronger relation-preserving construction: choose a least fixed-point region, transport its complement through a partial inverse, define a piecewise map, prove it surjective and injective, and discard the trivial relation witness.",
      "pages": [
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          "alt": "Non-exhaustive green and blue point fields show four representative injective strands in each direction, with a separate empty-case medallion above.",
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          "caption": "The selected theorem delegates to a stronger relation-preserving construction, with the empty case closed first.",
          "description": "schroeder_bernstein calls schroeder_bernstein_of_rel with the constantly true relation. Inside that helper, if β is empty then injectivity of f forces α empty and the empty equivalence supplies the result; the nonempty branch carries the fixed-point construction.",
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          "title": "Reduce to the stronger construction",
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        {
          "alt": "A shifted non-exhaustive source window surrounds a nested gold fixed-point island while a few injective sample strands connect the secondary target field.",
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          "caption": "A monotone complement-image operator selects the least source region on which the forward injection will be used.",
          "description": "The operator sends a source subset to the complement of g applied to the complement of its f-image. Mathlib takes F.lfp and uses F.map_lfp to identify the chosen region with its image under this operator; complement injectivity yields the corresponding identity for the outside region.",
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          "sha256": "sha256:998694ad25c725411ecf75d8c24f3631410479a4897ea43d2f35dca0b4f58752",
          "title": "Choose the least fixed-point region",
          "width": 1536,
          "equation": "F s = (g '' (f '' s)ᶜ)ᶜ; s = F.lfp",
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        {
          "alt": "Braided cobalt-and-gold strands carry the hatched source complement into the target complement beside a separate gold forward-image island.",
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          "caption": "The partial inverse of the backward injection sends the outside region onto precisely the target points missed by the forward branch.",
          "description": "leftInverse_invFun turns g' into a left inverse of g. Combining that fact with the fixed-point complement identity proves g' '' sᶜ = (f '' s)ᶜ, which is the exact coverage relation needed by the piecewise map.",
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          "title": "Transport the complement through a partial inverse",
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          "equation": "g' '' sᶜ = (f '' s)ᶜ",
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          "alt": "Gold strands from the selected source island and braided dashed strands from its hatched complement fill complementary regions of the blue target field.",
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          "caption": "The new map uses f on the least fixed-point region and the partial inverse of g outside it; the two images cover complementary target regions.",
          "description": "Mathlib defines h by Set.piecewise. range_piecewise splits its range into the f-image of s and the g'-image of sᶜ; substituting the complement identity and union_compl_self proves Surjective h.",
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          "title": "Define the piecewise map and cover the target",
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          "equation": "h = s.piecewise f g'; Function.Surjective h",
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          "alt": "Eight green points pair exactly with eight blue points through alternating solid-gold and cobalt-gold braided strands, each endpoint used once.",
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          "caption": "Branchwise injectivity and complement separation make the new surjective map one-to-one, yielding the required bijection.",
          "description": "injective_piecewise_iff uses hf on the selected region, the left-inverse law for g' on the complement, and the fact that an outside target cannot lie in f '' s. The helper returns h with both properties, and schroeder_bernstein specializes its relation to True and keeps the bijectivity witness.",
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          "sha256": "sha256:72ac4cce26a7eb67b2ca8e8d875a3f1d200b38bf4d842d955f6719f54fe861f0",
          "title": "Prove one-to-one and conclude",
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      "title": "How two injections produce a new bijection"
    },
    "whyLandmark": "Schröder–Bernstein turns two cardinal inequalities into an equality of cardinality. It is fundamental to comparing infinite sets without first constructing a bijection directly."
  },
  "nonClaims": [
    "Proof Atlas did not originate the Schröder–Bernstein theorem or Mathlib's declaration.",
    "The direct upstream theorem, not a local wrapper, is the canonical mathematical object.",
    "A historical or future review cannot be inferred from the build evidence alone."
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  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.schroeder-bernstein.v001",
  "statusAxes": {
    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "Current public presentation reviewed",
    "upstreamIndexed": "Pinned source bytes verified locally"
  },
  "summary": "Injections in both directions between two types imply a bijection between them.",
  "targetId": "target.library.mathlib.schroeder-bernstein.v001",
  "title": "Schröder–Bernstein Theorem",
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