Theorem isof1o 5488

Description: An isomorphism is a one-to-one onto function. (Contributed by set.mm contributors, 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-iso 4796	. 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 446	1 ⊢ (H Isom R, S (A, B) → H:A–1-1-onto→B)

Syntax hints:   → wi 4   ↔ wb 176  ∀wral 2614   class class class wbr 4639  –1-1-onto→wf1o 4780   ‘cfv 4781   Isom wiso 4782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360  df-iso 4796

This theorem is referenced by:  isomin  5496  isoini  5497  isoini2  5498