Theorem catcoppccl 16758

Description: The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
catcoppccl.c	|- C = (CatCat ` U )
catcoppccl.b	|- B = ( Base ` C )
catcoppccl.o	|- O = (oppCat ` X )
catcoppccl.1	|- ( ph -> U e. WUni )
catcoppccl.2	|- ( ph -> om e. U )
catcoppccl.3	|- ( ph -> X e. B )

Assertion

Ref	Expression
catcoppccl	|- ( ph -> O e. B )

Proof of Theorem catcoppccl

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

Step	Hyp	Ref	Expression
1		catcoppccl.3	. . . . 5 |- ( ph -> X e. B )
2		eqid 2622	. . . . . 6 |- ( Base ` X ) = ( Base ` X )
3		eqid 2622	. . . . . 6 |- ( Hom ` X ) = ( Hom ` X )
4		eqid 2622	. . . . . 6 |- (comp ` X ) = (comp ` X )
5		catcoppccl.o	. . . . . 6 |- O = (oppCat ` X )
6	2, 3, 4, 5	oppcval 16373	. . . . 5 |- ( X e. B -> O = ( ( X sSet <. ( Hom ` ndx ) , tpos ( Hom ` X ) >. ) sSet <. (comp ` ndx ) , ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) ) >. ) )
7	1, 6	syl 17	. . . 4 |- ( ph -> O = ( ( X sSet <. ( Hom ` ndx ) , tpos ( Hom ` X ) >. ) sSet <. (comp ` ndx ) , ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) ) >. ) )
8		catcoppccl.1	. . . . 5 |- ( ph -> U e. WUni )
9		inss1 3833	. . . . . . 7 |- ( U i^i Cat ) C_ U
10		catcoppccl.c	. . . . . . . . 9 |- C = (CatCat ` U )
11		catcoppccl.b	. . . . . . . . 9 |- B = ( Base ` C )
12	10, 11, 8	catcbas 16747	. . . . . . . 8 |- ( ph -> B = ( U i^i Cat ) )
13	1, 12	eleqtrd 2703	. . . . . . 7 |- ( ph -> X e. ( U i^i Cat ) )
14	9, 13	sseldi 3601	. . . . . 6 |- ( ph -> X e. U )
15		df-hom 15966	. . . . . . . 8 |- Hom = Slot ; 1 4
16		catcoppccl.2	. . . . . . . . 9 |- ( ph -> om e. U )
17	8, 16	wunndx 15878	. . . . . . . 8 |- ( ph -> ndx e. U )
18	15, 8, 17	wunstr 15881	. . . . . . 7 |- ( ph -> ( Hom ` ndx ) e. U )
19	15, 8, 14	wunstr 15881	. . . . . . . 8 |- ( ph -> ( Hom ` X ) e. U )
20	8, 19	wuntpos 9556	. . . . . . 7 |- ( ph -> tpos ( Hom ` X ) e. U )
21	8, 18, 20	wunop 9544	. . . . . 6 |- ( ph -> <. ( Hom ` ndx ) , tpos ( Hom ` X ) >. e. U )
22	8, 14, 21	wunsets 15900	. . . . 5 |- ( ph -> ( X sSet <. ( Hom ` ndx ) , tpos ( Hom ` X ) >. ) e. U )
23		df-cco 15967	. . . . . . 7 |- comp = Slot ; 1 5
24	23, 8, 17	wunstr 15881	. . . . . 6 |- ( ph -> (comp ` ndx ) e. U )
25		df-base 15863	. . . . . . . . . 10 |- Base = Slot 1
26	25, 8, 14	wunstr 15881	. . . . . . . . 9 |- ( ph -> ( Base ` X ) e. U )
27	8, 26, 26	wunxp 9546	. . . . . . . 8 |- ( ph -> ( ( Base ` X ) X. ( Base ` X ) ) e. U )
28	8, 27, 26	wunxp 9546	. . . . . . 7 |- ( ph -> ( ( ( Base ` X ) X. ( Base ` X ) ) X. ( Base ` X ) ) e. U )
29	23, 8, 14	wunstr 15881	. . . . . . . . . . . . . 14 |- ( ph -> (comp ` X ) e. U )
30	8, 29	wunrn 9551	. . . . . . . . . . . . 13 |- ( ph -> ran (comp ` X ) e. U )
31	8, 30	wununi 9528	. . . . . . . . . . . 12 |- ( ph -> U. ran (comp ` X ) e. U )
32	8, 31	wundm 9550	. . . . . . . . . . 11 |- ( ph -> dom U. ran (comp ` X ) e. U )
33	8, 32	wuncnv 9552	. . . . . . . . . 10 |- ( ph -> `' dom U. ran (comp ` X ) e. U )
34	8	wun0 9540	. . . . . . . . . . 11 |- ( ph -> (/) e. U )
35	8, 34	wunsn 9538	. . . . . . . . . 10 |- ( ph -> { (/) } e. U )
36	8, 33, 35	wunun 9532	. . . . . . . . 9 |- ( ph -> ( `' dom U. ran (comp ` X ) u. { (/) } ) e. U )
37	8, 31	wunrn 9551	. . . . . . . . 9 |- ( ph -> ran U. ran (comp ` X ) e. U )
38	8, 36, 37	wunxp 9546	. . . . . . . 8 |- ( ph -> ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) e. U )
39	8, 38	wunpw 9529	. . . . . . 7 |- ( ph -> ~P ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) e. U )
40		tposssxp 7356	. . . . . . . . . . . 12 |- tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ ( ( `' dom ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) u. { (/) } ) X. ran ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) )
41		ovssunirn 6681	. . . . . . . . . . . . . . 15 |- ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ U. ran (comp ` X )
42		dmss 5323	. . . . . . . . . . . . . . 15 |- ( ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ U. ran (comp ` X ) -> dom ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ dom U. ran (comp ` X ) )
43	41, 42	ax-mp 5	. . . . . . . . . . . . . 14 |- dom ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ dom U. ran (comp ` X )
44		cnvss 5294	. . . . . . . . . . . . . 14 |- ( dom ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ dom U. ran (comp ` X ) -> `' dom ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ `' dom U. ran (comp ` X ) )
45		unss1 3782	. . . . . . . . . . . . . 14 |- ( `' dom ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ `' dom U. ran (comp ` X ) -> ( `' dom ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) u. { (/) } ) C_ ( `' dom U. ran (comp ` X ) u. { (/) } ) )
46	43, 44, 45	mp2b 10	. . . . . . . . . . . . 13 |- ( `' dom ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) u. { (/) } ) C_ ( `' dom U. ran (comp ` X ) u. { (/) } )
47		rnss 5354	. . . . . . . . . . . . . 14 |- ( ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ U. ran (comp ` X ) -> ran ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ ran U. ran (comp ` X ) )
48	41, 47	ax-mp 5	. . . . . . . . . . . . 13 |- ran ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ ran U. ran (comp ` X )
49		xpss12 5225	. . . . . . . . . . . . 13 |- ( ( ( `' dom ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) u. { (/) } ) C_ ( `' dom U. ran (comp ` X ) u. { (/) } ) /\ ran ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ ran U. ran (comp ` X ) ) -> ( ( `' dom ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) u. { (/) } ) X. ran ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) ) C_ ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) )
50	46, 48, 49	mp2an 708	. . . . . . . . . . . 12 |- ( ( `' dom ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) u. { (/) } ) X. ran ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) ) C_ ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) )
51	40, 50	sstri 3612	. . . . . . . . . . 11 |- tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) )
52		elpw2g 4827	. . . . . . . . . . . 12 |- ( ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) e. U -> (tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) e. ~P ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) <-> tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) ) )
53	38, 52	syl 17	. . . . . . . . . . 11 |- ( ph -> (tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) e. ~P ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) <-> tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) ) )
54	51, 53	mpbiri 248	. . . . . . . . . 10 |- ( ph -> tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) e. ~P ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) )
55	54	ralrimivw 2967	. . . . . . . . 9 |- ( ph -> A. y e. ( Base ` X )tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) e. ~P ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) )
56	55	ralrimivw 2967	. . . . . . . 8 |- ( ph -> A. x e. ( ( Base ` X ) X. ( Base ` X ) ) A. y e. ( Base ` X )tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) e. ~P ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) )
57		eqid 2622	. . . . . . . . 9 |- ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) ) = ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) )
58	57	fmpt2 7237	. . . . . . . 8 |- ( A. x e. ( ( Base ` X ) X. ( Base ` X ) ) A. y e. ( Base ` X )tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) e. ~P ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) <-> ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) ) : ( ( ( Base ` X ) X. ( Base ` X ) ) X. ( Base ` X ) ) --> ~P ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) )
59	56, 58	sylib 208	. . . . . . 7 |- ( ph -> ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) ) : ( ( ( Base ` X ) X. ( Base ` X ) ) X. ( Base ` X ) ) --> ~P ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) )
60	8, 28, 39, 59	wunf 9549	. . . . . 6 |- ( ph -> ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) ) e. U )
61	8, 24, 60	wunop 9544	. . . . 5 |- ( ph -> <. (comp ` ndx ) , ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) ) >. e. U )
62	8, 22, 61	wunsets 15900	. . . 4 |- ( ph -> ( ( X sSet <. ( Hom ` ndx ) , tpos ( Hom ` X ) >. ) sSet <. (comp ` ndx ) , ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) ) >. ) e. U )
63	7, 62	eqeltrd 2701	. . 3 |- ( ph -> O e. U )
64		inss2 3834	. . . . 5 |- ( U i^i Cat ) C_ Cat
65	64, 13	sseldi 3601	. . . 4 |- ( ph -> X e. Cat )
66	5	oppccat 16382	. . . 4 |- ( X e. Cat -> O e. Cat )
67	65, 66	syl 17	. . 3 |- ( ph -> O e. Cat )
68	63, 67	elind 3798	. 2 |- ( ph -> O e. ( U i^i Cat ) )
69	68, 12	eleqtrrd 2704	1 |- ( ph -> O e. B )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   U.cuni 4436   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   1stc1st 7166   2ndc2nd 7167  tpos ctpos 7351  WUnicwun 9522   1c1 9937   4c4 11072   5c5 11073  ;cdc 11493   ndxcnx 15854   sSet csts 15855   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325  oppCatcoppc 16371  CatCatccatc 16744

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-inf2 8538  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-int 4476  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-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-wun 9524  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805  df-ltp 9807  df-enr 9877  df-nr 9878  df-c 9942  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-uz 11688  df-fz 12327  df-struct 15859  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-catc 16745

This theorem is referenced by: (None)