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 /\ A.x e. A A.y e. 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  A.wral 2615   class class class wbr 4640  -1-1-onto->wf1o 4781  ` cfv 4782   Isom wiso 4783

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 4797

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