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Theorem isof1o 5447
 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 4911 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
21simplbi 259 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wral 2306   class class class wbr 3764  –1-1-onto→wf1o 4901  ‘cfv 4902   Isom wiso 4903 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99 This theorem depends on definitions:  df-bi 110  df-isom 4911 This theorem is referenced by:  isocnv2  5452  isores1  5454  isoini  5457  isoini2  5458  isoselem  5459  isose  5460  isopolem  5461  isosolem  5463  smoiso  5917  ordiso2  6357
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