Metamath Proof Explorer

Theorem cofu2

Description: Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017)

		Ref	Expression
	Hypotheses	cofuval.b	\|- B = ( Base ` C )
		cofuval.f	\|- ( ph -> F e. ( C Func D ) )
		cofuval.g	\|- ( ph -> G e. ( D Func E ) )
		cofu2nd.x	\|- ( ph -> X e. B )
		cofu2nd.y	\|- ( ph -> Y e. B )
		cofu2.h	\|- H = ( Hom ` C )
		cofu2.y	\|- ( ph -> R e. ( X H Y ) )
	Assertion	cofu2	\|- ( ph -> ( ( X ( 2nd ` ( G o.func F ) ) Y ) ` R ) = ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) ` ( ( X ( 2nd ` F ) Y ) ` R ) ) )

Proof

Step	Hyp	Ref	Expression
1		cofuval.b	\|- B = ( Base ` C )
2		cofuval.f	\|- ( ph -> F e. ( C Func D ) )
3		cofuval.g	\|- ( ph -> G e. ( D Func E ) )
4		cofu2nd.x	\|- ( ph -> X e. B )
5		cofu2nd.y	\|- ( ph -> Y e. B )
6		cofu2.h	\|- H = ( Hom ` C )
7		cofu2.y	\|- ( ph -> R e. ( X H Y ) )
8	1 2 3 4 5	cofu2nd	\|- ( ph -> ( X ( 2nd ` ( G o.func F ) ) Y ) = ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) o. ( X ( 2nd ` F ) Y ) ) )
9	8	fveq1d	\|- ( ph -> ( ( X ( 2nd ` ( G o.func F ) ) Y ) ` R ) = ( ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) o. ( X ( 2nd ` F ) Y ) ) ` R ) )
10		eqid	\|- ( Hom ` D ) = ( Hom ` D )
11		relfunc	\|- Rel ( C Func D )
12		1st2ndbr	\|- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
13	11 2 12	sylancr	\|- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
14	1 6 10 13 4 5	funcf2	\|- ( ph -> ( X ( 2nd ` F ) Y ) : ( X H Y ) --> ( ( ( 1st ` F ) ` X ) ( Hom ` D ) ( ( 1st ` F ) ` Y ) ) )
15		fvco3	\|- ( ( ( X ( 2nd ` F ) Y ) : ( X H Y ) --> ( ( ( 1st ` F ) ` X ) ( Hom ` D ) ( ( 1st ` F ) ` Y ) ) /\ R e. ( X H Y ) ) -> ( ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) o. ( X ( 2nd ` F ) Y ) ) ` R ) = ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) ` ( ( X ( 2nd ` F ) Y ) ` R ) ) )
16	14 7 15	syl2anc	\|- ( ph -> ( ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) o. ( X ( 2nd ` F ) Y ) ) ` R ) = ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) ` ( ( X ( 2nd ` F ) Y ) ` R ) ) )
17	9 16	eqtrd	\|- ( ph -> ( ( X ( 2nd ` ( G o.func F ) ) Y ) ` R ) = ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) ` ( ( X ( 2nd ` F ) Y ) ` R ) ) )