Theorem 2oppccomf 16385

Description: The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 16397. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypothesis

Ref	Expression
oppcbas.1	|- O = (oppCat ` C )

Assertion

Ref	Expression
2oppccomf	|- (compf ` C ) = (compf ` (oppCat ` O ) )

Proof of Theorem 2oppccomf

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

Step	Hyp	Ref	Expression
1		oppcbas.1	. . . . . . . . 9 |- O = (oppCat ` C )
2		eqid 2622	. . . . . . . . 9 |- ( Base ` C ) = ( Base ` C )
3	1, 2	oppcbas 16378	. . . . . . . 8 |- ( Base ` C ) = ( Base ` O )
4		eqid 2622	. . . . . . . 8 |- (comp ` O ) = (comp ` O )
5		eqid 2622	. . . . . . . 8 |- (oppCat ` O ) = (oppCat ` O )
6		simpr1 1067	. . . . . . . 8 |- ( ( T. /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) ) -> x e. ( Base ` C ) )
7		simpr2 1068	. . . . . . . 8 |- ( ( T. /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) ) -> y e. ( Base ` C ) )
8		simpr3 1069	. . . . . . . 8 |- ( ( T. /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) ) -> z e. ( Base ` C ) )
9	3, 4, 5, 6, 7, 8	oppcco 16377	. . . . . . 7 |- ( ( T. /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) ) -> ( g ( <. x , y >. (comp ` (oppCat ` O ) ) z ) f ) = ( f ( <. z , y >. (comp ` O ) x ) g ) )
10		eqid 2622	. . . . . . . 8 |- (comp ` C ) = (comp ` C )
11	2, 10, 1, 8, 7, 6	oppcco 16377	. . . . . . 7 |- ( ( T. /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) ) -> ( f ( <. z , y >. (comp ` O ) x ) g ) = ( g ( <. x , y >. (comp ` C ) z ) f ) )
12	9, 11	eqtr2d 2657	. . . . . 6 |- ( ( T. /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) ) -> ( g ( <. x , y >. (comp ` C ) z ) f ) = ( g ( <. x , y >. (comp ` (oppCat ` O ) ) z ) f ) )
13	12	ralrimivw 2967	. . . . 5 |- ( ( T. /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) ) -> A. g e. ( y ( Hom ` C ) z ) ( g ( <. x , y >. (comp ` C ) z ) f ) = ( g ( <. x , y >. (comp ` (oppCat ` O ) ) z ) f ) )
14	13	ralrimivw 2967	. . . 4 |- ( ( T. /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) ) -> A. f e. ( x ( Hom ` C ) y ) A. g e. ( y ( Hom ` C ) z ) ( g ( <. x , y >. (comp ` C ) z ) f ) = ( g ( <. x , y >. (comp ` (oppCat ` O ) ) z ) f ) )
15	14	ralrimivvva 2972	. . 3 |- ( T. -> A. x e. ( Base ` C ) A. y e. ( Base ` C ) A. z e. ( Base ` C ) A. f e. ( x ( Hom ` C ) y ) A. g e. ( y ( Hom ` C ) z ) ( g ( <. x , y >. (comp ` C ) z ) f ) = ( g ( <. x , y >. (comp ` (oppCat ` O ) ) z ) f ) )
16		eqid 2622	. . . 4 |- (comp ` (oppCat ` O ) ) = (comp ` (oppCat ` O ) )
17		eqid 2622	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
18		eqidd 2623	. . . 4 |- ( T. -> ( Base ` C ) = ( Base ` C ) )
19	1, 2	2oppcbas 16383	. . . . 5 |- ( Base ` C ) = ( Base ` (oppCat ` O ) )
20	19	a1i 11	. . . 4 |- ( T. -> ( Base ` C ) = ( Base ` (oppCat ` O ) ) )
21	1	2oppchomf 16384	. . . . 5 |- ( Hom f ` C ) = ( Hom f ` (oppCat ` O ) )
22	21	a1i 11	. . . 4 |- ( T. -> ( Hom f ` C ) = ( Hom f ` (oppCat ` O ) ) )
23	10, 16, 17, 18, 20, 22	comfeq 16366	. . 3 |- ( T. -> ( (compf ` C ) = (compf ` (oppCat ` O ) ) <-> A. x e. ( Base ` C ) A. y e. ( Base ` C ) A. z e. ( Base ` C ) A. f e. ( x ( Hom ` C ) y ) A. g e. ( y ( Hom ` C ) z ) ( g ( <. x , y >. (comp ` C ) z ) f ) = ( g ( <. x , y >. (comp ` (oppCat ` O ) ) z ) f ) ) )
24	15, 23	mpbird 247	. 2 |- ( T. -> (compf ` C ) = (compf ` (oppCat ` O ) ) )
25	24	trud 1493	1 |- (compf ` C ) = (compf ` (oppCat ` O ) )

Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   e. wcel 1990   A.wral 2912   <.cop 4183   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Hom _f chomf 16327  comp_fccomf 16328  oppCatcoppc 16371

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-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-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-homf 16331  df-comf 16332  df-oppc 16372

This theorem is referenced by:  oppcepi  16399  oppchofcl  16900  oppcyon  16909  oyoncl  16910