Theorem cofu2 14916

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 14915	. . 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 5802	. 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 2454	. . . 4 |- ( Hom ` D ) = ( Hom ` D )
10		relfunc 14892	. . . . 5 |- Rel ( C Func D )
11		1st2ndbr 6734	. . . . 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 14898	. . 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 5878	. . 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 2495	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 1370   e. wcel 1758   class class class wbr 4401   o. ccom 4953   Rel wrel 4954   -->wf 5523   ` cfv 5527  (class class class)co 6201   1stc1st 6686   2ndc2nd 6687   Basecbs 14293   Hom chom 14369   Func cfunc 14884   o._func ccofu 14886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-map 7327  df-ixp 7375  df-func 14888  df-cofu 14890

This theorem is referenced by:  cofucl  14918  1st2ndprf  15136  uncf2  15167  yonedalem22  15208