Theorem funcco 16531

Description: A functor maps composition in the source category to composition in the target. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
funcco.b	|- B = ( Base ` D )
funcco.h	|- H = ( Hom ` D )
funcco.o	|- .x. = (comp ` D )
funcco.O	|- O = (comp ` E )
funcco.f	|- ( ph -> F ( D Func E ) G )
funcco.x	|- ( ph -> X e. B )
funcco.y	|- ( ph -> Y e. B )
funcco.z	|- ( ph -> Z e. B )
funcco.m	|- ( ph -> M e. ( X H Y ) )
funcco.n	|- ( ph -> N e. ( Y H Z ) )

Assertion

Ref	Expression
funcco	|- ( ph -> ( ( X G Z ) ` ( N ( <. X , Y >. .x. Z ) M ) ) = ( ( ( Y G Z ) ` N ) ( <. ( F ` X ) , ( F ` Y ) >. O ( F ` Z ) ) ( ( X G Y ) ` M ) ) )

Proof of Theorem funcco

Dummy variables m n x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funcco.f	. . . 4 |- ( ph -> F ( D Func E ) G )
2		funcco.b	. . . . 5 |- B = ( Base ` D )
3		eqid 2622	. . . . 5 |- ( Base ` E ) = ( Base ` E )
4		funcco.h	. . . . 5 |- H = ( Hom ` D )
5		eqid 2622	. . . . 5 |- ( Hom ` E ) = ( Hom ` E )
6		eqid 2622	. . . . 5 |- ( Id ` D ) = ( Id ` D )
7		eqid 2622	. . . . 5 |- ( Id ` E ) = ( Id ` E )
8		funcco.o	. . . . 5 |- .x. = (comp ` D )
9		funcco.O	. . . . 5 |- O = (comp ` E )
10		df-br 4654	. . . . . . . 8 |- ( F ( D Func E ) G <-> <. F , G >. e. ( D Func E ) )
11	1, 10	sylib 208	. . . . . . 7 |- ( ph -> <. F , G >. e. ( D Func E ) )
12		funcrcl 16523	. . . . . . 7 |- ( <. F , G >. e. ( D Func E ) -> ( D e. Cat /\ E e. Cat ) )
13	11, 12	syl 17	. . . . . 6 |- ( ph -> ( D e. Cat /\ E e. Cat ) )
14	13	simpld 475	. . . . 5 |- ( ph -> D e. Cat )
15	13	simprd 479	. . . . 5 |- ( ph -> E e. Cat )
16	2, 3, 4, 5, 6, 7, 8, 9, 14, 15	isfunc 16524	. . . 4 |- ( ph -> ( F ( D Func E ) G <-> ( F : B --> ( Base ` E ) /\ G e. X_ z e. ( B X. B ) ( ( ( F ` ( 1st ` z ) ) ( Hom ` E ) ( F ` ( 2nd ` z ) ) ) ^m ( H ` z ) ) /\ A. x e. B ( ( ( x G x ) ` ( ( Id ` D ) ` x ) ) = ( ( Id ` E ) ` ( F ` x ) ) /\ A. y e. B A. z e. B A. m e. ( x H y ) A. n e. ( y H z ) ( ( x G z ) ` ( n ( <. x , y >. .x. z ) m ) ) = ( ( ( y G z ) ` n ) ( <. ( F ` x ) , ( F ` y ) >. O ( F ` z ) ) ( ( x G y ) ` m ) ) ) ) ) )
17	1, 16	mpbid 222	. . 3 |- ( ph -> ( F : B --> ( Base ` E ) /\ G e. X_ z e. ( B X. B ) ( ( ( F ` ( 1st ` z ) ) ( Hom ` E ) ( F ` ( 2nd ` z ) ) ) ^m ( H ` z ) ) /\ A. x e. B ( ( ( x G x ) ` ( ( Id ` D ) ` x ) ) = ( ( Id ` E ) ` ( F ` x ) ) /\ A. y e. B A. z e. B A. m e. ( x H y ) A. n e. ( y H z ) ( ( x G z ) ` ( n ( <. x , y >. .x. z ) m ) ) = ( ( ( y G z ) ` n ) ( <. ( F ` x ) , ( F ` y ) >. O ( F ` z ) ) ( ( x G y ) ` m ) ) ) ) )
18	17	simp3d 1075	. 2 |- ( ph -> A. x e. B ( ( ( x G x ) ` ( ( Id ` D ) ` x ) ) = ( ( Id ` E ) ` ( F ` x ) ) /\ A. y e. B A. z e. B A. m e. ( x H y ) A. n e. ( y H z ) ( ( x G z ) ` ( n ( <. x , y >. .x. z ) m ) ) = ( ( ( y G z ) ` n ) ( <. ( F ` x ) , ( F ` y ) >. O ( F ` z ) ) ( ( x G y ) ` m ) ) ) )
19		funcco.x	. . 3 |- ( ph -> X e. B )
20		funcco.y	. . . . . 6 |- ( ph -> Y e. B )
21	20	adantr 481	. . . . 5 |- ( ( ph /\ x = X ) -> Y e. B )
22		funcco.z	. . . . . . 7 |- ( ph -> Z e. B )
23	22	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ x = X ) /\ y = Y ) -> Z e. B )
24		funcco.m	. . . . . . . . 9 |- ( ph -> M e. ( X H Y ) )
25	24	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> M e. ( X H Y ) )
26		simpllr 799	. . . . . . . . 9 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> x = X )
27		simplr 792	. . . . . . . . 9 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> y = Y )
28	26, 27	oveq12d 6668	. . . . . . . 8 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> ( x H y ) = ( X H Y ) )
29	25, 28	eleqtrrd 2704	. . . . . . 7 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> M e. ( x H y ) )
30		funcco.n	. . . . . . . . . 10 |- ( ph -> N e. ( Y H Z ) )
31	30	ad4antr 768	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) -> N e. ( Y H Z ) )
32		simpllr 799	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) -> y = Y )
33		simplr 792	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) -> z = Z )
34	32, 33	oveq12d 6668	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) -> ( y H z ) = ( Y H Z ) )
35	31, 34	eleqtrrd 2704	. . . . . . . 8 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) -> N e. ( y H z ) )
36		simp-5r 809	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> x = X )
37		simpllr 799	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> z = Z )
38	36, 37	oveq12d 6668	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> ( x G z ) = ( X G Z ) )
39		simp-4r 807	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> y = Y )
40	36, 39	opeq12d 4410	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> <. x , y >. = <. X , Y >. )
41	40, 37	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> ( <. x , y >. .x. z ) = ( <. X , Y >. .x. Z ) )
42		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> n = N )
43		simplr 792	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> m = M )
44	41, 42, 43	oveq123d 6671	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> ( n ( <. x , y >. .x. z ) m ) = ( N ( <. X , Y >. .x. Z ) M ) )
45	38, 44	fveq12d 6197	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> ( ( x G z ) ` ( n ( <. x , y >. .x. z ) m ) ) = ( ( X G Z ) ` ( N ( <. X , Y >. .x. Z ) M ) ) )
46	36	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> ( F ` x ) = ( F ` X ) )
47	39	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> ( F ` y ) = ( F ` Y ) )
48	46, 47	opeq12d 4410	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> <. ( F ` x ) , ( F ` y ) >. = <. ( F ` X ) , ( F ` Y ) >. )
49	37	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> ( F ` z ) = ( F ` Z ) )
50	48, 49	oveq12d 6668	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> ( <. ( F ` x ) , ( F ` y ) >. O ( F ` z ) ) = ( <. ( F ` X ) , ( F ` Y ) >. O ( F ` Z ) ) )
51	39, 37	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> ( y G z ) = ( Y G Z ) )
52	51, 42	fveq12d 6197	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> ( ( y G z ) ` n ) = ( ( Y G Z ) ` N ) )
53	36, 39	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> ( x G y ) = ( X G Y ) )
54	53, 43	fveq12d 6197	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> ( ( x G y ) ` m ) = ( ( X G Y ) ` M ) )
55	50, 52, 54	oveq123d 6671	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> ( ( ( y G z ) ` n ) ( <. ( F ` x ) , ( F ` y ) >. O ( F ` z ) ) ( ( x G y ) ` m ) ) = ( ( ( Y G Z ) ` N ) ( <. ( F ` X ) , ( F ` Y ) >. O ( F ` Z ) ) ( ( X G Y ) ` M ) ) )
56	45, 55	eqeq12d 2637	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) /\ n = N ) -> ( ( ( x G z ) ` ( n ( <. x , y >. .x. z ) m ) ) = ( ( ( y G z ) ` n ) ( <. ( F ` x ) , ( F ` y ) >. O ( F ` z ) ) ( ( x G y ) ` m ) ) <-> ( ( X G Z ) ` ( N ( <. X , Y >. .x. Z ) M ) ) = ( ( ( Y G Z ) ` N ) ( <. ( F ` X ) , ( F ` Y ) >. O ( F ` Z ) ) ( ( X G Y ) ` M ) ) ) )
57	35, 56	rspcdv 3312	. . . . . . 7 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ m = M ) -> ( A. n e. ( y H z ) ( ( x G z ) ` ( n ( <. x , y >. .x. z ) m ) ) = ( ( ( y G z ) ` n ) ( <. ( F ` x ) , ( F ` y ) >. O ( F ` z ) ) ( ( x G y ) ` m ) ) -> ( ( X G Z ) ` ( N ( <. X , Y >. .x. Z ) M ) ) = ( ( ( Y G Z ) ` N ) ( <. ( F ` X ) , ( F ` Y ) >. O ( F ` Z ) ) ( ( X G Y ) ` M ) ) ) )
58	29, 57	rspcimdv 3310	. . . . . 6 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> ( A. m e. ( x H y ) A. n e. ( y H z ) ( ( x G z ) ` ( n ( <. x , y >. .x. z ) m ) ) = ( ( ( y G z ) ` n ) ( <. ( F ` x ) , ( F ` y ) >. O ( F ` z ) ) ( ( x G y ) ` m ) ) -> ( ( X G Z ) ` ( N ( <. X , Y >. .x. Z ) M ) ) = ( ( ( Y G Z ) ` N ) ( <. ( F ` X ) , ( F ` Y ) >. O ( F ` Z ) ) ( ( X G Y ) ` M ) ) ) )
59	23, 58	rspcimdv 3310	. . . . 5 |- ( ( ( ph /\ x = X ) /\ y = Y ) -> ( A. z e. B A. m e. ( x H y ) A. n e. ( y H z ) ( ( x G z ) ` ( n ( <. x , y >. .x. z ) m ) ) = ( ( ( y G z ) ` n ) ( <. ( F ` x ) , ( F ` y ) >. O ( F ` z ) ) ( ( x G y ) ` m ) ) -> ( ( X G Z ) ` ( N ( <. X , Y >. .x. Z ) M ) ) = ( ( ( Y G Z ) ` N ) ( <. ( F ` X ) , ( F ` Y ) >. O ( F ` Z ) ) ( ( X G Y ) ` M ) ) ) )
60	21, 59	rspcimdv 3310	. . . 4 |- ( ( ph /\ x = X ) -> ( A. y e. B A. z e. B A. m e. ( x H y ) A. n e. ( y H z ) ( ( x G z ) ` ( n ( <. x , y >. .x. z ) m ) ) = ( ( ( y G z ) ` n ) ( <. ( F ` x ) , ( F ` y ) >. O ( F ` z ) ) ( ( x G y ) ` m ) ) -> ( ( X G Z ) ` ( N ( <. X , Y >. .x. Z ) M ) ) = ( ( ( Y G Z ) ` N ) ( <. ( F ` X ) , ( F ` Y ) >. O ( F ` Z ) ) ( ( X G Y ) ` M ) ) ) )
61	60	adantld 483	. . 3 |- ( ( ph /\ x = X ) -> ( ( ( ( x G x ) ` ( ( Id ` D ) ` x ) ) = ( ( Id ` E ) ` ( F ` x ) ) /\ A. y e. B A. z e. B A. m e. ( x H y ) A. n e. ( y H z ) ( ( x G z ) ` ( n ( <. x , y >. .x. z ) m ) ) = ( ( ( y G z ) ` n ) ( <. ( F ` x ) , ( F ` y ) >. O ( F ` z ) ) ( ( x G y ) ` m ) ) ) -> ( ( X G Z ) ` ( N ( <. X , Y >. .x. Z ) M ) ) = ( ( ( Y G Z ) ` N ) ( <. ( F ` X ) , ( F ` Y ) >. O ( F ` Z ) ) ( ( X G Y ) ` M ) ) ) )
62	19, 61	rspcimdv 3310	. 2 |- ( ph -> ( A. x e. B ( ( ( x G x ) ` ( ( Id ` D ) ` x ) ) = ( ( Id ` E ) ` ( F ` x ) ) /\ A. y e. B A. z e. B A. m e. ( x H y ) A. n e. ( y H z ) ( ( x G z ) ` ( n ( <. x , y >. .x. z ) m ) ) = ( ( ( y G z ) ` n ) ( <. ( F ` x ) , ( F ` y ) >. O ( F ` z ) ) ( ( x G y ) ` m ) ) ) -> ( ( X G Z ) ` ( N ( <. X , Y >. .x. Z ) M ) ) = ( ( ( Y G Z ) ` N ) ( <. ( F ` X ) , ( F ` Y ) >. O ( F ` Z ) ) ( ( X G Y ) ` M ) ) ) )
63	18, 62	mpd 15	1 |- ( ph -> ( ( X G Z ) ` ( N ( <. X , Y >. .x. Z ) M ) ) = ( ( ( Y G Z ) ` N ) ( <. ( F ` X ) , ( F ` Y ) >. O ( F ` Z ) ) ( ( X G Y ) ` M ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   class class class wbr 4653   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   X_cixp 7908   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326   Func cfunc 16514

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-map 7859  df-ixp 7909  df-func 16518

This theorem is referenced by:  funcsect  16532  funcoppc  16535  cofucl  16548  funcres  16556  fthsect  16585  fthmon  16587  catcisolem  16756  prfcl  16843  evlfcllem  16861  curf1cl  16868  curf2cl  16871  curfcl  16872  uncfcurf  16879  yonedalem4c  16917