Theorem invinv

Description: The inverse of the inverse of an isomorphism is itself. Proposition 3.14(1) of Adamek p. 29. (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 )
	invinv.f	|- ( ph -> F e. ( X I Y ) )
Assertion	invinv	|- ( ph -> ( ( Y N X ) ` ( ( X N Y ) ` F ) ) = F )

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		invinv.f	|- ( ph -> F e. ( X I Y ) )
8	1 2 3 4 5	invsym2	|- ( ph -> `' ( X N Y ) = ( Y N X ) )
9	8	fveq1d	|- ( ph -> ( `' ( X N Y ) ` ( ( X N Y ) ` F ) ) = ( ( Y N X ) ` ( ( X N Y ) ` F ) ) )
10	1 2 3 4 5 6	invf1o	|- ( ph -> ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) )
11		f1ocnvfv1	|- ( ( ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) /\ F e. ( X I Y ) ) -> ( `' ( X N Y ) ` ( ( X N Y ) ` F ) ) = F )
12	10 7 11	syl2anc	|- ( ph -> ( `' ( X N Y ) ` ( ( X N Y ) ` F ) ) = F )
13	9 12	eqtr3d	|- ( ph -> ( ( Y N X ) ` ( ( X N Y ) ` F ) ) = F )