Theorem isof1o 5447

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 4911	. 2 |- ( H Isom R , S ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) )
2	1	simplbi 259	1 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )

Syntax hints:   -> wi 4   <-> wb 98   A.wral 2306   class class class wbr 3764   -1-1-onto->wf1o 4901   ` cfv 4902   Isom wiso 4903

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99

This theorem depends on definitions:  df-bi 110  df-isom 4911

This theorem is referenced by:  isocnv2  5452  isores1  5454  isoini  5457  isoini2  5458  isoselem  5459  isose  5460  isopolem  5461  isosolem  5463  smoiso  5917  ordiso2  6357