Theorem catcoppccl 17362

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	⊢ (𝜑 → 𝑈 ∈ WUni)
catcoppccl.2	⊢ (𝜑 → ω ∈ 𝑈)
catcoppccl.3	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
catcoppccl	⊢ (𝜑 → 𝑂 ∈ 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 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		eqid 2821	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
3		eqid 2821	. . . . . 6 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
4		eqid 2821	. . . . . 6 ⊢ (comp‘𝑋) = (comp‘𝑋)
5		catcoppccl.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝑋)
6	2, 3, 4, 5	oppcval 16977	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑂 = ((𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) sSet ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)))⟩))
7	1, 6	syl 17	. . . 4 ⊢ (𝜑 → 𝑂 = ((𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) sSet ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)))⟩))
8		catcoppccl.1	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ WUni)
9		catcoppccl.c	. . . . . . . . 9 ⊢ 𝐶 = (CatCat‘𝑈)
10		catcoppccl.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
11	9, 10, 8	catcbas 17351	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
12	1, 11	eleqtrd 2915	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
13	12	elin1d 4174	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
14		df-hom 16583	. . . . . . . 8 ⊢ Hom = Slot ;14
15		catcoppccl.2	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ 𝑈)
16	8, 15	wunndx 16498	. . . . . . . 8 ⊢ (𝜑 → ndx ∈ 𝑈)
17	14, 8, 16	wunstr 16501	. . . . . . 7 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
18	14, 8, 13	wunstr 16501	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
19	8, 18	wuntpos 10150	. . . . . . 7 ⊢ (𝜑 → tpos (Hom ‘𝑋) ∈ 𝑈)
20	8, 17, 19	wunop 10138	. . . . . 6 ⊢ (𝜑 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩ ∈ 𝑈)
21	8, 13, 20	wunsets 16518	. . . . 5 ⊢ (𝜑 → (𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) ∈ 𝑈)
22		df-cco 16584	. . . . . . 7 ⊢ comp = Slot ;15
23	22, 8, 16	wunstr 16501	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
24		df-base 16483	. . . . . . . . . 10 ⊢ Base = Slot 1
25	24, 8, 13	wunstr 16501	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
26	8, 25, 25	wunxp 10140	. . . . . . . 8 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑋)) ∈ 𝑈)
27	8, 26, 25	wunxp 10140	. . . . . . 7 ⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋)) ∈ 𝑈)
28	22, 8, 13	wunstr 16501	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
29	8, 28	wunrn 10145	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
30	8, 29	wununi 10122	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈)
31	8, 30	wundm 10144	. . . . . . . . . . 11 ⊢ (𝜑 → dom ∪ ran (comp‘𝑋) ∈ 𝑈)
32	8, 31	wuncnv 10146	. . . . . . . . . 10 ⊢ (𝜑 → ◡dom ∪ ran (comp‘𝑋) ∈ 𝑈)
33	8	wun0 10134	. . . . . . . . . . 11 ⊢ (𝜑 → ∅ ∈ 𝑈)
34	8, 33	wunsn 10132	. . . . . . . . . 10 ⊢ (𝜑 → {∅} ∈ 𝑈)
35	8, 32, 34	wunun 10126	. . . . . . . . 9 ⊢ (𝜑 → (◡dom ∪ ran (comp‘𝑋) ∪ {∅}) ∈ 𝑈)
36	8, 30	wunrn 10145	. . . . . . . . 9 ⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈)
37	8, 35, 36	wunxp 10140	. . . . . . . 8 ⊢ (𝜑 → ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
38	8, 37	wunpw 10123	. . . . . . 7 ⊢ (𝜑 → 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
39		tposssxp 7890	. . . . . . . . . . . 12 ⊢ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ((◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∪ {∅}) × ran (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)))
40		ovssunirn 7186	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ∪ ran (comp‘𝑋)
41		dmss 5765	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ∪ ran (comp‘𝑋) → dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ dom ∪ ran (comp‘𝑋))
42	40, 41	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ dom ∪ ran (comp‘𝑋)
43		cnvss 5737	. . . . . . . . . . . . . 14 ⊢ (dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ dom ∪ ran (comp‘𝑋) → ◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ◡dom ∪ ran (comp‘𝑋))
44		unss1 4154	. . . . . . . . . . . . . 14 ⊢ (◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ◡dom ∪ ran (comp‘𝑋) → (◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∪ {∅}) ⊆ (◡dom ∪ ran (comp‘𝑋) ∪ {∅}))
45	42, 43, 44	mp2b 10	. . . . . . . . . . . . 13 ⊢ (◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∪ {∅}) ⊆ (◡dom ∪ ran (comp‘𝑋) ∪ {∅})
46	40	rnssi 5804	. . . . . . . . . . . . 13 ⊢ ran (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ran ∪ ran (comp‘𝑋)
47		xpss12 5564	. . . . . . . . . . . . 13 ⊢ (((◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∪ {∅}) ⊆ (◡dom ∪ ran (comp‘𝑋) ∪ {∅}) ∧ ran (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ran ∪ ran (comp‘𝑋)) → ((◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∪ {∅}) × ran (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥))) ⊆ ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
48	45, 46, 47	mp2an 690	. . . . . . . . . . . 12 ⊢ ((◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∪ {∅}) × ran (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥))) ⊆ ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))
49	39, 48	sstri 3975	. . . . . . . . . . 11 ⊢ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))
50		elpw2g 5239	. . . . . . . . . . . 12 ⊢ (((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈 → (tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∈ 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ↔ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))))
51	37, 50	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∈ 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ↔ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))))
52	49, 51	mpbiri 260	. . . . . . . . . 10 ⊢ (𝜑 → tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∈ 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
53	52	ralrimivw 3183	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∈ 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
54	53	ralrimivw 3183	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋))∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∈ 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
55		eqid 2821	. . . . . . . . 9 ⊢ (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥))) = (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)))
56	55	fmpo 7760	. . . . . . . 8 ⊢ (∀𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋))∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∈ 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ↔ (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥))):(((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋))⟶𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
57	54, 56	sylib 220	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥))):(((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋))⟶𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
58	8, 27, 38, 57	wunf 10143	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥))) ∈ 𝑈)
59	8, 23, 58	wunop 10138	. . . . 5 ⊢ (𝜑 → ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)))⟩ ∈ 𝑈)
60	8, 21, 59	wunsets 16518	. . . 4 ⊢ (𝜑 → ((𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) sSet ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)))⟩) ∈ 𝑈)
61	7, 60	eqeltrd 2913	. . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑈)
62	12	elin2d 4175	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Cat)
63	5	oppccat 16986	. . . 4 ⊢ (𝑋 ∈ Cat → 𝑂 ∈ Cat)
64	62, 63	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
65	61, 64	elind 4170	. 2 ⊢ (𝜑 → 𝑂 ∈ (𝑈 ∩ Cat))
66	65, 11	eleqtrrd 2916	1 ⊢ (𝜑 → 𝑂 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110  ∀wral 3138   ∪ cun 3933   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538  {csn 4560  ⟨cop 4566  ∪ cuni 4831   × cxp 5547  ^◡ccnv 5548  dom cdm 5549  ran crn 5550  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150   ∈ cmpo 7152  ωcom 7574  1^st c1st 7681  2^nd c2nd 7682  tpos ctpos 7885  WUnicwun 10116  1c1 10532  4c4 11688  5c5 11689  ;cdc 12092  ndxcnx 16474   sSet csts 16475  Basecbs 16477  Hom chom 16570  compcco 16571  Catccat 16929  oppCatcoppc 16975  CatCatccatc 17348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-tpos 7886  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-omul 8101  df-er 8283  df-ec 8285  df-qs 8289  df-map 8402  df-pm 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-wun 10118  df-ni 10288  df-pli 10289  df-mi 10290  df-lti 10291  df-plpq 10324  df-mpq 10325  df-ltpq 10326  df-enq 10327  df-nq 10328  df-erq 10329  df-plq 10330  df-mq 10331  df-1nq 10332  df-rq 10333  df-ltnq 10334  df-np 10397  df-plp 10399  df-ltp 10401  df-enr 10471  df-nr 10472  df-c 10537  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12887  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-hom 16583  df-cco 16584  df-cat 16933  df-cid 16934  df-oppc 16976  df-catc 17349

This theorem is referenced by: (None)