Theorem cic

Description: Objects X and Y in a category are isomorphic provided that there is an isomorphism f : X --> Y , see definition 3.15 of Adamek p. 29. (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 )
Assertion	cic	|- ( ph -> ( X ( ~=c ` C ) Y <-> E. f f e. ( X I 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	1 2 3 4 5	brcic	|- ( ph -> ( X ( ~=c ` C ) Y <-> ( X I Y ) =/= (/) ) )
7		n0	|- ( ( X I Y ) =/= (/) <-> E. f f e. ( X I Y ) )
8	6 7	bitrdi	|- ( ph -> ( X ( ~=c ` C ) Y <-> E. f f e. ( X I Y ) ) )