Theorem cofu2 15102

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

Hypotheses

Ref	Expression
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

Ref	Expression
cofu2	|- ( ph -> ( ( X ( 2nd ` ( G o.func F ) ) Y ) ` R ) = ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) ` ( ( X ( 2nd ` F ) Y ) ` R ) ) )

Proof of Theorem cofu2

Step	Hyp	Ref	Expression
1		cofuval.b	. . . 4 |- B = ( Base ` C )
2		cofuval.f	. . . 4 |- ( ph -> F e. ( C Func D ) )
3		cofuval.g	. . . 4 |- ( ph -> G e. ( D Func E ) )
4		cofu2nd.x	. . . 4 |- ( ph -> X e. B )
5		cofu2nd.y	. . . 4 |- ( ph -> Y e. B )
6	1, 2, 3, 4, 5	cofu2nd 15101	. . 3 |- ( ph -> ( X ( 2nd ` ( G o.func F ) ) Y ) = ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) o. ( X ( 2nd ` F ) Y ) ) )
7	6	fveq1d 5859	. 2 |- ( ph -> ( ( X ( 2nd ` ( G o.func F ) ) Y ) ` R ) = ( ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) o. ( X ( 2nd ` F ) Y ) ) ` R ) )
8		cofu2.h	. . . 4 |- H = ( Hom ` C )
9		eqid 2460	. . . 4 |- ( Hom ` D ) = ( Hom ` D )
10		relfunc 15078	. . . . 5 |- Rel ( C Func D )
11		1st2ndbr 6823	. . . . 5 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
12	10, 2, 11	sylancr 663	. . . 4 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
13	1, 8, 9, 12, 4, 5	funcf2 15084	. . 3 |- ( ph -> ( X ( 2nd ` F ) Y ) : ( X H Y ) --> ( ( ( 1st ` F ) ` X ) ( Hom ` D ) ( ( 1st ` F ) ` Y ) ) )
14		cofu2.y	. . 3 |- ( ph -> R e. ( X H Y ) )
15		fvco3 5935	. . 3 |- ( ( ( 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	13, 14, 15	syl2anc 661	. 2 |- ( 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	7, 16	eqtrd 2501	1 |- ( ph -> ( ( X ( 2nd ` ( G o.func F ) ) Y ) ` R ) = ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) ` ( ( X ( 2nd ` F ) Y ) ` R ) ) )

Syntax hints:   -> wi 4   = wceq 1374   e. wcel 1762   class class class wbr 4440   o. ccom 4996   Rel wrel 4997   -->wf 5575   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   Basecbs 14479   Hom chom 14555   Func cfunc 15070   o._func ccofu 15072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-map 7412  df-ixp 7460  df-func 15074  df-cofu 15076

This theorem is referenced by:  cofucl  15104  1st2ndprf  15322  uncf2  15353  yonedalem22  15394