Theorem funcoppc 16535

Description: A functor on categories yields a functor on the opposite categories (in the same direction), see definition 3.41 of [Adamek] p. 39. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
funcoppc.o	|- O = (oppCat ` C )
funcoppc.p	|- P = (oppCat ` D )
funcoppc.f	|- ( ph -> F ( C Func D ) G )

Assertion

Ref	Expression
funcoppc	|- ( ph -> F ( O Func P )tpos G )

Proof of Theorem funcoppc

Dummy variables f g x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funcoppc.o	. . 3 |- O = (oppCat ` C )
2		eqid 2622	. . 3 |- ( Base ` C ) = ( Base ` C )
3	1, 2	oppcbas 16378	. 2 |- ( Base ` C ) = ( Base ` O )
4		funcoppc.p	. . 3 |- P = (oppCat ` D )
5		eqid 2622	. . 3 |- ( Base ` D ) = ( Base ` D )
6	4, 5	oppcbas 16378	. 2 |- ( Base ` D ) = ( Base ` P )
7		eqid 2622	. 2 |- ( Hom ` O ) = ( Hom ` O )
8		eqid 2622	. 2 |- ( Hom ` P ) = ( Hom ` P )
9		eqid 2622	. 2 |- ( Id ` O ) = ( Id ` O )
10		eqid 2622	. 2 |- ( Id ` P ) = ( Id ` P )
11		eqid 2622	. 2 |- (comp ` O ) = (comp ` O )
12		eqid 2622	. 2 |- (comp ` P ) = (comp ` P )
13		funcoppc.f	. . . . . 6 |- ( ph -> F ( C Func D ) G )
14		df-br 4654	. . . . . 6 |- ( F ( C Func D ) G <-> <. F , G >. e. ( C Func D ) )
15	13, 14	sylib 208	. . . . 5 |- ( ph -> <. F , G >. e. ( C Func D ) )
16		funcrcl 16523	. . . . 5 |- ( <. F , G >. e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
17	15, 16	syl 17	. . . 4 |- ( ph -> ( C e. Cat /\ D e. Cat ) )
18	17	simpld 475	. . 3 |- ( ph -> C e. Cat )
19	1	oppccat 16382	. . 3 |- ( C e. Cat -> O e. Cat )
20	18, 19	syl 17	. 2 |- ( ph -> O e. Cat )
21	17	simprd 479	. . 3 |- ( ph -> D e. Cat )
22	4	oppccat 16382	. . 3 |- ( D e. Cat -> P e. Cat )
23	21, 22	syl 17	. 2 |- ( ph -> P e. Cat )
24	2, 5, 13	funcf1 16526	. 2 |- ( ph -> F : ( Base ` C ) --> ( Base ` D ) )
25	2, 13	funcfn2 16529	. . 3 |- ( ph -> G Fn ( ( Base ` C ) X. ( Base ` C ) ) )
26		tposfn 7381	. . 3 |- ( G Fn ( ( Base ` C ) X. ( Base ` C ) ) -> tpos G Fn ( ( Base ` C ) X. ( Base ` C ) ) )
27	25, 26	syl 17	. 2 |- ( ph -> tpos G Fn ( ( Base ` C ) X. ( Base ` C ) ) )
28		eqid 2622	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
29		eqid 2622	. . . 4 |- ( Hom ` D ) = ( Hom ` D )
30	13	adantr 481	. . . 4 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> F ( C Func D ) G )
31		simprr 796	. . . 4 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> y e. ( Base ` C ) )
32		simprl 794	. . . 4 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> x e. ( Base ` C ) )
33	2, 28, 29, 30, 31, 32	funcf2 16528	. . 3 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( y G x ) : ( y ( Hom ` C ) x ) --> ( ( F ` y ) ( Hom ` D ) ( F ` x ) ) )
34		ovtpos 7367	. . . . 5 |- ( xtpos G y ) = ( y G x )
35	34	feq1i 6036	. . . 4 |- ( ( xtpos G y ) : ( x ( Hom ` O ) y ) --> ( ( F ` x ) ( Hom ` P ) ( F ` y ) ) <-> ( y G x ) : ( x ( Hom ` O ) y ) --> ( ( F ` x ) ( Hom ` P ) ( F ` y ) ) )
36	28, 1	oppchom 16375	. . . . 5 |- ( x ( Hom ` O ) y ) = ( y ( Hom ` C ) x )
37	29, 4	oppchom 16375	. . . . 5 |- ( ( F ` x ) ( Hom ` P ) ( F ` y ) ) = ( ( F ` y ) ( Hom ` D ) ( F ` x ) )
38	36, 37	feq23i 6039	. . . 4 |- ( ( y G x ) : ( x ( Hom ` O ) y ) --> ( ( F ` x ) ( Hom ` P ) ( F ` y ) ) <-> ( y G x ) : ( y ( Hom ` C ) x ) --> ( ( F ` y ) ( Hom ` D ) ( F ` x ) ) )
39	35, 38	bitri 264	. . 3 |- ( ( xtpos G y ) : ( x ( Hom ` O ) y ) --> ( ( F ` x ) ( Hom ` P ) ( F ` y ) ) <-> ( y G x ) : ( y ( Hom ` C ) x ) --> ( ( F ` y ) ( Hom ` D ) ( F ` x ) ) )
40	33, 39	sylibr 224	. 2 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( xtpos G y ) : ( x ( Hom ` O ) y ) --> ( ( F ` x ) ( Hom ` P ) ( F ` y ) ) )
41		eqid 2622	. . . 4 |- ( Id ` C ) = ( Id ` C )
42		eqid 2622	. . . 4 |- ( Id ` D ) = ( Id ` D )
43	13	adantr 481	. . . 4 |- ( ( ph /\ x e. ( Base ` C ) ) -> F ( C Func D ) G )
44		simpr 477	. . . 4 |- ( ( ph /\ x e. ( Base ` C ) ) -> x e. ( Base ` C ) )
45	2, 41, 42, 43, 44	funcid 16530	. . 3 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( x G x ) ` ( ( Id ` C ) ` x ) ) = ( ( Id ` D ) ` ( F ` x ) ) )
46		ovtpos 7367	. . . . 5 |- ( xtpos G x ) = ( x G x )
47	46	a1i 11	. . . 4 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( xtpos G x ) = ( x G x ) )
48	1, 41	oppcid 16381	. . . . . . 7 |- ( C e. Cat -> ( Id ` O ) = ( Id ` C ) )
49	18, 48	syl 17	. . . . . 6 |- ( ph -> ( Id ` O ) = ( Id ` C ) )
50	49	adantr 481	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( Id ` O ) = ( Id ` C ) )
51	50	fveq1d 6193	. . . 4 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( Id ` O ) ` x ) = ( ( Id ` C ) ` x ) )
52	47, 51	fveq12d 6197	. . 3 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( xtpos G x ) ` ( ( Id ` O ) ` x ) ) = ( ( x G x ) ` ( ( Id ` C ) ` x ) ) )
53	4, 42	oppcid 16381	. . . . . 6 |- ( D e. Cat -> ( Id ` P ) = ( Id ` D ) )
54	21, 53	syl 17	. . . . 5 |- ( ph -> ( Id ` P ) = ( Id ` D ) )
55	54	adantr 481	. . . 4 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( Id ` P ) = ( Id ` D ) )
56	55	fveq1d 6193	. . 3 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( Id ` P ) ` ( F ` x ) ) = ( ( Id ` D ) ` ( F ` x ) ) )
57	45, 52, 56	3eqtr4d 2666	. 2 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( xtpos G x ) ` ( ( Id ` O ) ` x ) ) = ( ( Id ` P ) ` ( F ` x ) ) )
58		eqid 2622	. . . . 5 |- (comp ` C ) = (comp ` C )
59		eqid 2622	. . . . 5 |- (comp ` D ) = (comp ` D )
60	13	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> F ( C Func D ) G )
61		simp23 1096	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> z e. ( Base ` C ) )
62		simp22 1095	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> y e. ( Base ` C ) )
63		simp21 1094	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> x e. ( Base ` C ) )
64		simp3r 1090	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> g e. ( y ( Hom ` O ) z ) )
65	28, 1	oppchom 16375	. . . . . 6 |- ( y ( Hom ` O ) z ) = ( z ( Hom ` C ) y )
66	64, 65	syl6eleq 2711	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> g e. ( z ( Hom ` C ) y ) )
67		simp3l 1089	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> f e. ( x ( Hom ` O ) y ) )
68	67, 36	syl6eleq 2711	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> f e. ( y ( Hom ` C ) x ) )
69	2, 28, 58, 59, 60, 61, 62, 63, 66, 68	funcco 16531	. . . 4 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> ( ( z G x ) ` ( f ( <. z , y >. (comp ` C ) x ) g ) ) = ( ( ( y G x ) ` f ) ( <. ( F ` z ) , ( F ` y ) >. (comp ` D ) ( F ` x ) ) ( ( z G y ) ` g ) ) )
70	2, 58, 1, 63, 62, 61	oppcco 16377	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> ( g ( <. x , y >. (comp ` O ) z ) f ) = ( f ( <. z , y >. (comp ` C ) x ) g ) )
71	70	fveq2d 6195	. . . 4 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> ( ( z G x ) ` ( g ( <. x , y >. (comp ` O ) z ) f ) ) = ( ( z G x ) ` ( f ( <. z , y >. (comp ` C ) x ) g ) ) )
72	24	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> F : ( Base ` C ) --> ( Base ` D ) )
73	72, 63	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> ( F ` x ) e. ( Base ` D ) )
74	72, 62	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> ( F ` y ) e. ( Base ` D ) )
75	72, 61	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> ( F ` z ) e. ( Base ` D ) )
76	5, 59, 4, 73, 74, 75	oppcco 16377	. . . 4 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> ( ( ( z G y ) ` g ) ( <. ( F ` x ) , ( F ` y ) >. (comp ` P ) ( F ` z ) ) ( ( y G x ) ` f ) ) = ( ( ( y G x ) ` f ) ( <. ( F ` z ) , ( F ` y ) >. (comp ` D ) ( F ` x ) ) ( ( z G y ) ` g ) ) )
77	69, 71, 76	3eqtr4d 2666	. . 3 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> ( ( z G x ) ` ( g ( <. x , y >. (comp ` O ) z ) f ) ) = ( ( ( z G y ) ` g ) ( <. ( F ` x ) , ( F ` y ) >. (comp ` P ) ( F ` z ) ) ( ( y G x ) ` f ) ) )
78		ovtpos 7367	. . . 4 |- ( xtpos G z ) = ( z G x )
79	78	fveq1i 6192	. . 3 |- ( ( xtpos G z ) ` ( g ( <. x , y >. (comp ` O ) z ) f ) ) = ( ( z G x ) ` ( g ( <. x , y >. (comp ` O ) z ) f ) )
80		ovtpos 7367	. . . . 5 |- ( ytpos G z ) = ( z G y )
81	80	fveq1i 6192	. . . 4 |- ( ( ytpos G z ) ` g ) = ( ( z G y ) ` g )
82	34	fveq1i 6192	. . . 4 |- ( ( xtpos G y ) ` f ) = ( ( y G x ) ` f )
83	81, 82	oveq12i 6662	. . 3 |- ( ( ( ytpos G z ) ` g ) ( <. ( F ` x ) , ( F ` y ) >. (comp ` P ) ( F ` z ) ) ( ( xtpos G y ) ` f ) ) = ( ( ( z G y ) ` g ) ( <. ( F ` x ) , ( F ` y ) >. (comp ` P ) ( F ` z ) ) ( ( y G x ) ` f ) )
84	77, 79, 83	3eqtr4g 2681	. 2 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` O ) y ) /\ g e. ( y ( Hom ` O ) z ) ) ) -> ( ( xtpos G z ) ` ( g ( <. x , y >. (comp ` O ) z ) f ) ) = ( ( ( ytpos G z ) ` g ) ( <. ( F ` x ) , ( F ` y ) >. (comp ` P ) ( F ` z ) ) ( ( xtpos G y ) ` f ) ) )
85	3, 6, 7, 8, 9, 10, 11, 12, 20, 23, 24, 27, 40, 57, 84	isfuncd 16525	1 |- ( ph -> F ( O Func P )tpos G )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650  tpos ctpos 7351   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326  oppCatcoppc 16371   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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-oppc 16372  df-func 16518

This theorem is referenced by:  fulloppc  16582  fthoppc  16583  yonedalem1  16912  yonedalem21  16913  yonedalem22  16918