Theorem catcoppccl 18091

Description: The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)

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 2727	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
3		eqid 2727	. . . . . 6 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
4		eqid 2727	. . . . . 6 ⊢ (comp‘𝑋) = (comp‘𝑋)
5		catcoppccl.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝑋)
6	2, 3, 4, 5	oppcval 17678	. . . . 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	. . . . . . 7 ⊢ 𝐶 = (CatCat‘𝑈)
10		catcoppccl.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
11	9, 10, 8, 1	catcbascl 18086	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
12		homid 17378	. . . . . . . 8 ⊢ Hom = Slot (Hom ‘ndx)
13		catcoppccl.2	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ 𝑈)
14	8, 13	wunndx 17149	. . . . . . . 8 ⊢ (𝜑 → ndx ∈ 𝑈)
15	12, 8, 14	wunstr 17142	. . . . . . 7 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
16	9, 10, 8, 1	catchomcl 18089	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
17	8, 16	wuntpos 10743	. . . . . . 7 ⊢ (𝜑 → tpos (Hom ‘𝑋) ∈ 𝑈)
18	8, 15, 17	wunop 10731	. . . . . 6 ⊢ (𝜑 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩ ∈ 𝑈)
19	8, 11, 18	wunsets 17131	. . . . 5 ⊢ (𝜑 → (𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) ∈ 𝑈)
20		ccoid 17380	. . . . . . 7 ⊢ comp = Slot (comp‘ndx)
21	20, 8, 14	wunstr 17142	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
22	9, 10, 8, 1	catcbaselcl 18088	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
23	8, 22, 22	wunxp 10733	. . . . . . . 8 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑋)) ∈ 𝑈)
24	8, 23, 22	wunxp 10733	. . . . . . 7 ⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋)) ∈ 𝑈)
25	9, 10, 8, 1	catcccocl 18090	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
26	8, 25	wunrn 10738	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
27	8, 26	wununi 10715	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈)
28	8, 27	wundm 10737	. . . . . . . . . . 11 ⊢ (𝜑 → dom ∪ ran (comp‘𝑋) ∈ 𝑈)
29	8, 28	wuncnv 10739	. . . . . . . . . 10 ⊢ (𝜑 → ◡dom ∪ ran (comp‘𝑋) ∈ 𝑈)
30	8	wun0 10727	. . . . . . . . . . 11 ⊢ (𝜑 → ∅ ∈ 𝑈)
31	8, 30	wunsn 10725	. . . . . . . . . 10 ⊢ (𝜑 → {∅} ∈ 𝑈)
32	8, 29, 31	wunun 10719	. . . . . . . . 9 ⊢ (𝜑 → (◡dom ∪ ran (comp‘𝑋) ∪ {∅}) ∈ 𝑈)
33	8, 27	wunrn 10738	. . . . . . . . 9 ⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈)
34	8, 32, 33	wunxp 10733	. . . . . . . 8 ⊢ (𝜑 → ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
35	8, 34	wunpw 10716	. . . . . . 7 ⊢ (𝜑 → 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
36		tposssxp 8227	. . . . . . . . . . . 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)))
37		ovssunirn 7450	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ∪ ran (comp‘𝑋)
38		dmss 5899	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ∪ ran (comp‘𝑋) → dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ dom ∪ ran (comp‘𝑋))
39	37, 38	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ dom ∪ ran (comp‘𝑋)
40		cnvss 5869	. . . . . . . . . . . . . 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))
41		unss1 4175	. . . . . . . . . . . . . 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) ∪ {∅}))
42	39, 40, 41	mp2b 10	. . . . . . . . . . . . 13 ⊢ (◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∪ {∅}) ⊆ (◡dom ∪ ran (comp‘𝑋) ∪ {∅})
43	37	rnssi 5936	. . . . . . . . . . . . 13 ⊢ ran (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ran ∪ ran (comp‘𝑋)
44		xpss12 5687	. . . . . . . . . . . . 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)))
45	42, 43, 44	mp2an 691	. . . . . . . . . . . 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))
46	36, 45	sstri 3987	. . . . . . . . . . 11 ⊢ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))
47		elpw2g 5340	. . . . . . . . . . . 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))))
48	34, 47	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))))
49	46, 48	mpbiri 258	. . . . . . . . . 10 ⊢ (𝜑 → tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∈ 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
50	49	ralrimivw 3145	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∈ 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
51	50	ralrimivw 3145	. . . . . . . 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)))
52		eqid 2727	. . . . . . . . 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)))
53	52	fmpo 8064	. . . . . . . 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)))
54	51, 53	sylib 217	. . . . . . 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)))
55	8, 24, 35, 54	wunf 10736	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥))) ∈ 𝑈)
56	8, 21, 55	wunop 10731	. . . . 5 ⊢ (𝜑 → ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)))⟩ ∈ 𝑈)
57	8, 19, 56	wunsets 17131	. . . 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)
58	7, 57	eqeltrd 2828	. . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑈)
59	9, 10, 8	catcbas 18075	. . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
60	1, 59	eleqtrd 2830	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
61	60	elin2d 4195	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Cat)
62	5	oppccat 17689	. . . 4 ⊢ (𝑋 ∈ Cat → 𝑂 ∈ Cat)
63	61, 62	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
64	58, 63	elind 4190	. 2 ⊢ (𝜑 → 𝑂 ∈ (𝑈 ∩ Cat))
65	64, 59	eleqtrrd 2831	1 ⊢ (𝜑 → 𝑂 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1534   ∈ wcel 2099  ∀wral 3056   ∪ cun 3942   ∩ cin 3943   ⊆ wss 3944  ∅c0 4318  𝒫 cpw 4598  {csn 4624  ⟨cop 4630  ∪ cuni 4903   × cxp 5670  ^◡ccnv 5671  dom cdm 5672  ran crn 5673  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416  ωcom 7862  1^st c1st 7983  2^nd c2nd 7984  tpos ctpos 8222  WUnicwun 10709   sSet csts 17117  ndxcnx 17147  Basecbs 17165  Hom chom 17229  compcco 17230  Catccat 17629  oppCatcoppc 17676  CatCatccatc 18072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-inf2 9650  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-tpos 8223  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-oadd 8482  df-omul 8483  df-er 8716  df-ec 8718  df-qs 8722  df-map 8836  df-pm 8837  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-wun 10711  df-ni 10881  df-pli 10882  df-mi 10883  df-lti 10884  df-plpq 10917  df-mpq 10918  df-ltpq 10919  df-enq 10920  df-nq 10921  df-erq 10922  df-plq 10923  df-mq 10924  df-1nq 10925  df-rq 10926  df-ltnq 10927  df-np 10990  df-plp 10992  df-ltp 10994  df-enr 11064  df-nr 11065  df-c 11130  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-3 12292  df-4 12293  df-5 12294  df-6 12295  df-7 12296  df-8 12297  df-9 12298  df-n0 12489  df-z 12575  df-dec 12694  df-uz 12839  df-fz 13503  df-struct 17101  df-sets 17118  df-slot 17136  df-ndx 17148  df-base 17166  df-hom 17242  df-cco 17243  df-cat 17633  df-cid 17634  df-oppc 17677  df-catc 18073

This theorem is referenced by: (None)