Theorem brcici

Description: Prove that two objects are isomorphic by an explicit isomorphism. (Contributed by AV, 4-Apr-2020)

	Ref	Expression
Hypotheses	cic.i	|- I = ( Iso ` C )
	cic.b	|- B = ( Base ` C )
	cic.c	|- ( ph -> C e. Cat )
	cic.x	|- ( ph -> X e. B )
	cic.y	|- ( ph -> Y e. B )
	cic.f	|- ( ph -> F e. ( X I Y ) )
Assertion	brcici	|- ( ph -> X ( ~=c ` C ) Y )

Proof

Step	Hyp	Ref	Expression
1		cic.i	|- I = ( Iso ` C )
2		cic.b	|- B = ( Base ` C )
3		cic.c	|- ( ph -> C e. Cat )
4		cic.x	|- ( ph -> X e. B )
5		cic.y	|- ( ph -> Y e. B )
6		cic.f	|- ( ph -> F e. ( X I Y ) )
7		eleq1	|- ( f = F -> ( f e. ( X I Y ) <-> F e. ( X I Y ) ) )
8	7	spcegv	|- ( F e. ( X I Y ) -> ( F e. ( X I Y ) -> E. f f e. ( X I Y ) ) )
9	6 6 8	sylc	|- ( ph -> E. f f e. ( X I Y ) )
10	1 2 3 4 5	cic	|- ( ph -> ( X ( ~=c ` C ) Y <-> E. f f e. ( X I Y ) ) )
11	9 10	mpbird	|- ( ph -> X ( ~=c ` C ) Y )