Theorem catcoppccl 14997

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 2443	. . . . . 6 |- ( Base ` X ) = ( Base ` X )
3		eqid 2443	. . . . . 6 |- ( Hom ` X ) = ( Hom ` X )
4		eqid 2443	. . . . . 6 |- (comp ` X ) = (comp ` X )
5		catcoppccl.o	. . . . . 6 |- O = (oppCat ` X )
6	2, 3, 4, 5	oppcval 14673	. . . . 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 16	. . . 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 3591	. . . . . . 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 14986	. . . . . . . 8 |- ( ph -> B = ( U i^i Cat ) )
13	1, 12	eleqtrd 2519	. . . . . . 7 |- ( ph -> X e. ( U i^i Cat ) )
14	9, 13	sseldi 3375	. . . . . 6 |- ( ph -> X e. U )
15		df-hom 14283	. . . . . . . 8 |- Hom = Slot ; 1 4
16		catcoppccl.2	. . . . . . . . 9 |- ( ph -> om e. U )
17	8, 16	wunndx 14211	. . . . . . . 8 |- ( ph -> ndx e. U )
18	15, 8, 17	wunstr 14214	. . . . . . 7 |- ( ph -> ( Hom ` ndx ) e. U )
19	15, 8, 14	wunstr 14214	. . . . . . . 8 |- ( ph -> ( Hom ` X ) e. U )
20	8, 19	wuntpos 8922	. . . . . . 7 |- ( ph -> tpos ( Hom ` X ) e. U )
21	8, 18, 20	wunop 8910	. . . . . 6 |- ( ph -> <. ( Hom ` ndx ) , tpos ( Hom ` X ) >. e. U )
22	8, 14, 21	wunsets 14222	. . . . 5 |- ( ph -> ( X sSet <. ( Hom ` ndx ) , tpos ( Hom ` X ) >. ) e. U )
23		df-cco 14284	. . . . . . 7 |- comp = Slot ; 1 5
24	23, 8, 17	wunstr 14214	. . . . . 6 |- ( ph -> (comp ` ndx ) e. U )
25		df-base 14200	. . . . . . . . . 10 |- Base = Slot 1
26	25, 8, 14	wunstr 14214	. . . . . . . . 9 |- ( ph -> ( Base ` X ) e. U )
27	8, 26, 26	wunxp 8912	. . . . . . . 8 |- ( ph -> ( ( Base ` X ) X. ( Base ` X ) ) e. U )
28	8, 27, 26	wunxp 8912	. . . . . . 7 |- ( ph -> ( ( ( Base ` X ) X. ( Base ` X ) ) X. ( Base ` X ) ) e. U )
29	23, 8, 14	wunstr 14214	. . . . . . . . . . . . . 14 |- ( ph -> (comp ` X ) e. U )
30	8, 29	wunrn 8917	. . . . . . . . . . . . 13 |- ( ph -> ran (comp ` X ) e. U )
31	8, 30	wununi 8894	. . . . . . . . . . . 12 |- ( ph -> U. ran (comp ` X ) e. U )
32	8, 31	wundm 8916	. . . . . . . . . . 11 |- ( ph -> dom U. ran (comp ` X ) e. U )
33	8, 32	wuncnv 8918	. . . . . . . . . 10 |- ( ph -> `' dom U. ran (comp ` X ) e. U )
34	8	wun0 8906	. . . . . . . . . . 11 |- ( ph -> (/) e. U )
35	8, 34	wunsn 8904	. . . . . . . . . 10 |- ( ph -> { (/) } e. U )
36	8, 33, 35	wunun 8898	. . . . . . . . 9 |- ( ph -> ( `' dom U. ran (comp ` X ) u. { (/) } ) e. U )
37	8, 31	wunrn 8917	. . . . . . . . 9 |- ( ph -> ran U. ran (comp ` X ) e. U )
38	8, 36, 37	wunxp 8912	. . . . . . . 8 |- ( ph -> ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) e. U )
39	8, 38	wunpw 8895	. . . . . . 7 |- ( ph -> ~P ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) ) e. U )
40		tposssxp 6770	. . . . . . . . . . . 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 6138	. . . . . . . . . . . . . . 15 |- ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ U. ran (comp ` X )
42		dmss 5060	. . . . . . . . . . . . . . 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 5033	. . . . . . . . . . . . . 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 3546	. . . . . . . . . . . . . 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 5089	. . . . . . . . . . . . . 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 4966	. . . . . . . . . . . . 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 672	. . . . . . . . . . . 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 3386	. . . . . . . . . . 11 |- tpos ( <. y , ( 2nd ` x ) >. (comp ` X ) ( 1st ` x ) ) C_ ( ( `' dom U. ran (comp ` X ) u. { (/) } ) X. ran U. ran (comp ` X ) )
52		elpw2g 4476	. . . . . . . . . . . 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 16	. . . . . . . . . . 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 233	. . . . . . . . . 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 2821	. . . . . . . . 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 2821	. . . . . . . 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 2443	. . . . . . . . 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 6662	. . . . . . . 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 196	. . . . . . 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 8915	. . . . . 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 8910	. . . . 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 14222	. . . 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 2517	. . 3 |- ( ph -> O e. U )
64		inss2 3592	. . . . 5 |- ( U i^i Cat ) C_ Cat
65	64, 13	sseldi 3375	. . . 4 |- ( ph -> X e. Cat )
66	5	oppccat 14682	. . . 4 |- ( X e. Cat -> O e. Cat )
67	65, 66	syl 16	. . 3 |- ( ph -> O e. Cat )
68	63, 67	elind 3561	. 2 |- ( ph -> O e. ( U i^i Cat ) )
69	68, 12	eleqtrrd 2520	1 |- ( ph -> O e. B )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1369   e. wcel 1756   A.wral 2736   u. cun 3347   i^i cin 3348   C_ wss 3349   (/)c0 3658   ~Pcpw 3881   {csn 3898   <.cop 3904   U.cuni 4112   X. cxp 4859   `'ccnv 4860   dom cdm 4861   ran crn 4862   -->wf 5435   ` cfv 5439  (class class class)co 6112   e. cmpt2 6114   omcom 6497   1stc1st 6596   2ndc2nd 6597  tpos ctpos 6765  WUnicwun 8888   1c1 9304   4c4 10394   5c5 10395  ;cdc 10776   ndxcnx 14192   sSet csts 14193   Basecbs 14195   Hom chom 14270  compcco 14271   Catccat 14623  oppCatcoppc 14671  CatCatccatc 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-tpos 6766  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-omul 6946  df-er 7122  df-ec 7124  df-qs 7128  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-wun 8890  df-ni 9062  df-pli 9063  df-mi 9064  df-lti 9065  df-plpq 9098  df-mpq 9099  df-ltpq 9100  df-enq 9101  df-nq 9102  df-erq 9103  df-plq 9104  df-mq 9105  df-1nq 9106  df-rq 9107  df-ltnq 9108  df-np 9171  df-plp 9173  df-ltp 9175  df-enr 9247  df-nr 9248  df-c 9309  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-fz 11459  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-hom 14283  df-cco 14284  df-cat 14627  df-cid 14628  df-oppc 14672  df-catc 14984

This theorem is referenced by: (None)