Theorem isof1o 6003

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 5381	. 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 2528   class class class wbr 4125  –1-1-onto→wf1o 5371  ‘cfv 5372   Isom wiso 5373

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 5381

This theorem is referenced by:  isocnv2  6008  isores1  6010  isoini  6014  isoini2  6015  isoselem  6016  isose  6017  isopolem  6018  isosolem  6020  smoiso  6563  isotilem  7336  supisolem  7338  supisoex  7339  supisoti  7340  ordiso2  7365  leisorel  11267  zfz1isolemiso  11269  seq3coll  11272  summodclem2a  12126  prodmodclem2a  12321