Theorem invf1o

Description: The inverse relation is a bijection from isomorphisms to isomorphisms. This means that every isomorphism F e. ( X I Y ) has a unique inverse, denoted by ( ( InvC )F ) . Remark 3.12 of Adamek p. 28. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	invfval.b	|- B = ( Base ` C )
	invfval.n	|- N = ( Inv ` C )
	invfval.c	|- ( ph -> C e. Cat )
	invfval.x	|- ( ph -> X e. B )
	invfval.y	|- ( ph -> Y e. B )
	isoval.n	|- I = ( Iso ` C )
Assertion	invf1o	|- ( ph -> ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) )

Proof

Step	Hyp	Ref	Expression
1		invfval.b	|- B = ( Base ` C )
2		invfval.n	|- N = ( Inv ` C )
3		invfval.c	|- ( ph -> C e. Cat )
4		invfval.x	|- ( ph -> X e. B )
5		invfval.y	|- ( ph -> Y e. B )
6		isoval.n	|- I = ( Iso ` C )
7	1 2 3 4 5 6	invf	|- ( ph -> ( X N Y ) : ( X I Y ) --> ( Y I X ) )
8	7	ffnd	|- ( ph -> ( X N Y ) Fn ( X I Y ) )
9	1 2 3 5 4 6	invf	|- ( ph -> ( Y N X ) : ( Y I X ) --> ( X I Y ) )
10	9	ffnd	|- ( ph -> ( Y N X ) Fn ( Y I X ) )
11	1 2 3 4 5	invsym2	|- ( ph -> `' ( X N Y ) = ( Y N X ) )
12	11	fneq1d	|- ( ph -> ( `' ( X N Y ) Fn ( Y I X ) <-> ( Y N X ) Fn ( Y I X ) ) )
13	10 12	mpbird	|- ( ph -> `' ( X N Y ) Fn ( Y I X ) )
14		dff1o4	|- ( ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) <-> ( ( X N Y ) Fn ( X I Y ) /\ `' ( X N Y ) Fn ( Y I X ) ) )
15	8 13 14	sylanbrc	|- ( ph -> ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) )