Theorem cofu2 16546

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 16545	. . 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 6193	. 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 2622	. . . 4 |- ( Hom ` D ) = ( Hom ` D )
10		relfunc 16522	. . . . 5 |- Rel ( C Func D )
11		1st2ndbr 7217	. . . . 5 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
12	10, 2, 11	sylancr 695	. . . 4 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
13	1, 8, 9, 12, 4, 5	funcf2 16528	. . 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 6275	. . 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 693	. 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 2656	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 1483   e. wcel 1990   class class class wbr 4653   o. ccom 5118   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952   Func cfunc 16514   o._func ccofu 16516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-func 16518  df-cofu 16520

This theorem is referenced by:  cofucl  16548  1st2ndprf  16846  uncf2  16877  yonedalem22  16918