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Theorem isof1o 5475
Description: An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)
Assertion
Ref Expression
isof1o (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)

Proof of Theorem isof1o
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-isom 4939 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
21simplbi 263 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wral 2323   class class class wbr 3792  1-1-ontowf1o 4929  cfv 4930   Isom wiso 4931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103
This theorem depends on definitions:  df-bi 114  df-isom 4939
This theorem is referenced by:  isocnv2  5480  isores1  5482  isoini  5485  isoini2  5486  isoselem  5487  isose  5488  isopolem  5489  isosolem  5491  smoiso  5948  isotilem  6410  supisolem  6412  supisoex  6413  supisoti  6414  ordiso2  6415
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