Theorem isof1o 5467

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 4931	. 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 268	1 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )

Syntax hints:   -> wi 4   <-> wb 103   A.wral 2348   class class class wbr 3785   -1-1-onto->wf1o 4921   ` cfv 4922   Isom wiso 4923

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104

This theorem depends on definitions:  df-bi 115  df-isom 4931

This theorem is referenced by:  isocnv2  5472  isores1  5474  isoini  5477  isoini2  5478  isoselem  5479  isose  5480  isopolem  5481  isosolem  5483  smoiso  5940  isotilem  6419  supisolem  6421  supisoex  6422  supisoti  6423  ordiso2  6446