Theorem isof1o 5810

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:𝐴–1-1-onto→𝐵)

Proof of Theorem isof1o

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-isom 5227	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
2	1	simplbi 274	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 105  ∀wral 2455   class class class wbr 4005  –1-1-onto→wf1o 5217  ‘cfv 5218   Isom wiso 5219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This theorem depends on definitions:  df-bi 117  df-isom 5227

This theorem is referenced by:  isocnv2  5815  isores1  5817  isoini  5821  isoini2  5822  isoselem  5823  isose  5824  isopolem  5825  isosolem  5827  smoiso  6305  isotilem  7007  supisolem  7009  supisoex  7010  supisoti  7011  ordiso2  7036  leisorel  10819  zfz1isolemiso  10821  seq3coll  10824  summodclem2a  11391  prodmodclem2a  11586