Theorem cofu2 15499

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 15498	. . 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 5851	. 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 2402	. . . 4 |- ( Hom ` D ) = ( Hom ` D )
10		relfunc 15475	. . . . 5 |- Rel ( C Func D )
11		1st2ndbr 6833	. . . . 5 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
12	10, 2, 11	sylancr 661	. . . 4 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
13	1, 8, 9, 12, 4, 5	funcf2 15481	. . 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 5926	. . 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 659	. 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 2443	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 1405   e. wcel 1842   class class class wbr 4395   o. ccom 4827   Rel wrel 4828   -->wf 5565   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783   Basecbs 14841   Hom chom 14920   Func cfunc 15467   o._func ccofu 15469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-map 7459  df-ixp 7508  df-func 15471  df-cofu 15473

This theorem is referenced by:  cofucl  15501  1st2ndprf  15799  uncf2  15830  yonedalem22  15871