Theorem isof1o 5489

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 4797	. 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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wral 2615   class class class wbr 4640  –1-1-onto→wf1o 4781   ‘cfv 4782   Isom wiso 4783

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360  df-iso 4797

This theorem is used by:  isomin  5497  isoini  5498  isoini2  5499