Theorem catcoppccl 18026

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 2733	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
3		eqid 2733	. . . . . 6 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
4		eqid 2733	. . . . . 6 ⊢ (comp‘𝑋) = (comp‘𝑋)
5		catcoppccl.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝑋)
6	2, 3, 4, 5	oppcval 17621	. . . . 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 18021	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
12		homid 17318	. . . . . . . 8 ⊢ Hom = Slot (Hom ‘ndx)
13		catcoppccl.2	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ 𝑈)
14	8, 13	wunndx 17108	. . . . . . . 8 ⊢ (𝜑 → ndx ∈ 𝑈)
15	12, 8, 14	wunstr 17101	. . . . . . 7 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
16	9, 10, 8, 1	catchomcl 18024	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
17	8, 16	wuntpos 10632	. . . . . . 7 ⊢ (𝜑 → tpos (Hom ‘𝑋) ∈ 𝑈)
18	8, 15, 17	wunop 10620	. . . . . 6 ⊢ (𝜑 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩ ∈ 𝑈)
19	8, 11, 18	wunsets 17090	. . . . 5 ⊢ (𝜑 → (𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) ∈ 𝑈)
20		ccoid 17320	. . . . . . 7 ⊢ comp = Slot (comp‘ndx)
21	20, 8, 14	wunstr 17101	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
22	9, 10, 8, 1	catcbaselcl 18023	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
23	8, 22, 22	wunxp 10622	. . . . . . . 8 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑋)) ∈ 𝑈)
24	8, 23, 22	wunxp 10622	. . . . . . 7 ⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋)) ∈ 𝑈)
25	9, 10, 8, 1	catcccocl 18025	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
26	8, 25	wunrn 10627	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
27	8, 26	wununi 10604	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈)
28	8, 27	wundm 10626	. . . . . . . . . . 11 ⊢ (𝜑 → dom ∪ ran (comp‘𝑋) ∈ 𝑈)
29	8, 28	wuncnv 10628	. . . . . . . . . 10 ⊢ (𝜑 → ◡dom ∪ ran (comp‘𝑋) ∈ 𝑈)
30	8	wun0 10616	. . . . . . . . . . 11 ⊢ (𝜑 → ∅ ∈ 𝑈)
31	8, 30	wunsn 10614	. . . . . . . . . 10 ⊢ (𝜑 → {∅} ∈ 𝑈)
32	8, 29, 31	wunun 10608	. . . . . . . . 9 ⊢ (𝜑 → (◡dom ∪ ran (comp‘𝑋) ∪ {∅}) ∈ 𝑈)
33	8, 27	wunrn 10627	. . . . . . . . 9 ⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈)
34	8, 32, 33	wunxp 10622	. . . . . . . 8 ⊢ (𝜑 → ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
35	8, 34	wunpw 10605	. . . . . . 7 ⊢ (𝜑 → 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
36		tposssxp 8166	. . . . . . . . . . . 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 7388	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ∪ ran (comp‘𝑋)
38		dmss 5846	. . . . . . . . . . . . . . 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 5816	. . . . . . . . . . . . . 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 4134	. . . . . . . . . . . . . 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 5884	. . . . . . . . . . . . 13 ⊢ ran (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ran ∪ ran (comp‘𝑋)
44		xpss12 5634	. . . . . . . . . . . . 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 692	. . . . . . . . . . . 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 3940	. . . . . . . . . . 11 ⊢ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))
47		elpw2g 5273	. . . . . . . . . . . 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 3129	. . . . . . . . 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 3129	. . . . . . . 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 2733	. . . . . . . . 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 8006	. . . . . . . 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 218	. . . . . . 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 10625	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥))) ∈ 𝑈)
56	8, 21, 55	wunop 10620	. . . . 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 17090	. . . 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 2833	. . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑈)
59	9, 10, 8	catcbas 18010	. . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
60	1, 59	eleqtrd 2835	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
61	60	elin2d 4154	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Cat)
62	5	oppccat 17630	. . . 4 ⊢ (𝑋 ∈ Cat → 𝑂 ∈ Cat)
63	61, 62	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
64	58, 63	elind 4149	. 2 ⊢ (𝜑 → 𝑂 ∈ (𝑈 ∩ Cat))
65	64, 59	eleqtrrd 2836	1 ⊢ (𝜑 → 𝑂 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  ∀wral 3048   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4549  {csn 4575  ⟨cop 4581  ∪ cuni 4858   × cxp 5617  ^◡ccnv 5618  dom cdm 5619  ran crn 5620  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354  ωcom 7802  1^st c1st 7925  2^nd c2nd 7926  tpos ctpos 8161  WUnicwun 10598   sSet csts 17076  ndxcnx 17106  Basecbs 17122  Hom chom 17174  compcco 17175  Catccat 17572  oppCatcoppc 17619  CatCatccatc 18007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9538  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-tpos 8162  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-oadd 8395  df-omul 8396  df-er 8628  df-ec 8630  df-qs 8634  df-map 8758  df-pm 8759  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-wun 10600  df-ni 10770  df-pli 10771  df-mi 10772  df-lti 10773  df-plpq 10806  df-mpq 10807  df-ltpq 10808  df-enq 10809  df-nq 10810  df-erq 10811  df-plq 10812  df-mq 10813  df-1nq 10814  df-rq 10815  df-ltnq 10816  df-np 10879  df-plp 10881  df-ltp 10883  df-enr 10953  df-nr 10954  df-c 11019  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-9 12202  df-n0 12389  df-z 12476  df-dec 12595  df-uz 12739  df-fz 13410  df-struct 17060  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-hom 17187  df-cco 17188  df-cat 17576  df-cid 17577  df-oppc 17620  df-catc 18008

This theorem is referenced by: (None)