catcoppccl - Metamath Proof Explorer

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Theorem catcoppccl 18079

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 2726	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
3		eqid 2726	. . . . . 6 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
4		eqid 2726	. . . . . 6 ⊢ (comp‘𝑋) = (comp‘𝑋)
5		catcoppccl.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝑋)
6	2, 3, 4, 5	oppcval 17666	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑂 = ((𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) sSet ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘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, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘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 18074	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
12		homid 17366	. . . . . . . 8 ⊢ Hom = Slot (Hom ‘ndx)
13		catcoppccl.2	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ 𝑈)
14	8, 13	wunndx 17137	. . . . . . . 8 ⊢ (𝜑 → ndx ∈ 𝑈)
15	12, 8, 14	wunstr 17130	. . . . . . 7 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
16	9, 10, 8, 1	catchomcl 18077	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
17	8, 16	wuntpos 10731	. . . . . . 7 ⊢ (𝜑 → tpos (Hom ‘𝑋) ∈ 𝑈)
18	8, 15, 17	wunop 10719	. . . . . 6 ⊢ (𝜑 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩ ∈ 𝑈)
19	8, 11, 18	wunsets 17119	. . . . 5 ⊢ (𝜑 → (𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) ∈ 𝑈)
20		ccoid 17368	. . . . . . 7 ⊢ comp = Slot (comp‘ndx)
21	20, 8, 14	wunstr 17130	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
22	9, 10, 8, 1	catcbaselcl 18076	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
23	8, 22, 22	wunxp 10721	. . . . . . . 8 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑋)) ∈ 𝑈)
24	8, 23, 22	wunxp 10721	. . . . . . 7 ⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋)) ∈ 𝑈)
25	9, 10, 8, 1	catcccocl 18078	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
26	8, 25	wunrn 10726	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
27	8, 26	wununi 10703	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈)
28	8, 27	wundm 10725	. . . . . . . . . . 11 ⊢ (𝜑 → dom ∪ ran (comp‘𝑋) ∈ 𝑈)
29	8, 28	wuncnv 10727	. . . . . . . . . 10 ⊢ (𝜑 → ^◡dom ∪ ran (comp‘𝑋) ∈ 𝑈)
30	8	wun0 10715	. . . . . . . . . . 11 ⊢ (𝜑 → ∅ ∈ 𝑈)
31	8, 30	wunsn 10713	. . . . . . . . . 10 ⊢ (𝜑 → {∅} ∈ 𝑈)
32	8, 29, 31	wunun 10707	. . . . . . . . 9 ⊢ (𝜑 → (^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) ∈ 𝑈)
33	8, 27	wunrn 10726	. . . . . . . . 9 ⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈)
34	8, 32, 33	wunxp 10721	. . . . . . . 8 ⊢ (𝜑 → ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
35	8, 34	wunpw 10704	. . . . . . 7 ⊢ (𝜑 → 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
36		tposssxp 8216	. . . . . . . . . . . 12 ⊢ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ((^◡dom (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∪ {∅}) × ran (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)))
37		ovssunirn 7441	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ∪ ran (comp‘𝑋)
38		dmss 5896	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ∪ ran (comp‘𝑋) → dom (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ dom ∪ ran (comp‘𝑋))
39	37, 38	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ dom ∪ ran (comp‘𝑋)
40		cnvss 5866	. . . . . . . . . . . . . 14 ⊢ (dom (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ dom ∪ ran (comp‘𝑋) → ^◡dom (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ^◡dom ∪ ran (comp‘𝑋))
41		unss1 4174	. . . . . . . . . . . . . 14 ⊢ (^◡dom (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ^◡dom ∪ ran (comp‘𝑋) → (^◡dom (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∪ {∅}) ⊆ (^◡dom ∪ ran (comp‘𝑋) ∪ {∅}))
42	39, 40, 41	mp2b 10	. . . . . . . . . . . . 13 ⊢ (^◡dom (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∪ {∅}) ⊆ (^◡dom ∪ ran (comp‘𝑋) ∪ {∅})
43	37	rnssi 5933	. . . . . . . . . . . . 13 ⊢ ran (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ran ∪ ran (comp‘𝑋)
44		xpss12 5684	. . . . . . . . . . . . 13 ⊢ (((^◡dom (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∪ {∅}) ⊆ (^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) ∧ ran (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ran ∪ ran (comp‘𝑋)) → ((^◡dom (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∪ {∅}) × ran (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥))) ⊆ ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
45	42, 43, 44	mp2an 689	. . . . . . . . . . . 12 ⊢ ((^◡dom (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∪ {∅}) × ran (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥))) ⊆ ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))
46	36, 45	sstri 3986	. . . . . . . . . . 11 ⊢ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))
47		elpw2g 5337	. . . . . . . . . . . 12 ⊢ (((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈 → (tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∈ 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ↔ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))))
48	34, 47	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∈ 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ↔ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))))
49	46, 48	mpbiri 258	. . . . . . . . . 10 ⊢ (𝜑 → tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∈ 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
50	49	ralrimivw 3144	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∈ 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
51	50	ralrimivw 3144	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋))∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∈ 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
52		eqid 2726	. . . . . . . . 9 ⊢ (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥))) = (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)))
53	52	fmpo 8053	. . . . . . . 8 ⊢ (∀𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋))∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∈ 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ↔ (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘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, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥))):(((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋))⟶𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
55	8, 24, 35, 54	wunf 10724	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥))) ∈ 𝑈)
56	8, 21, 55	wunop 10719	. . . . 5 ⊢ (𝜑 → ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)))⟩ ∈ 𝑈)
57	8, 19, 56	wunsets 17119	. . . 4 ⊢ (𝜑 → ((𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) sSet ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)))⟩) ∈ 𝑈)
58	7, 57	eqeltrd 2827	. . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑈)
59	9, 10, 8	catcbas 18063	. . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
60	1, 59	eleqtrd 2829	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
61	60	elin2d 4194	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Cat)
62	5	oppccat 17677	. . . 4 ⊢ (𝑋 ∈ Cat → 𝑂 ∈ Cat)
63	61, 62	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
64	58, 63	elind 4189	. 2 ⊢ (𝜑 → 𝑂 ∈ (𝑈 ∩ Cat))
65	64, 59	eleqtrrd 2830	1 ⊢ (𝜑 → 𝑂 ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∪ cun 3941 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 𝒫 cpw 4597 {csn 4623 ⟨cop 4629 ∪ cuni 4902 × cxp 5667 ^◡ccnv 5668 dom cdm 5669 ran crn 5670 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 ωcom 7852 1^st c1st 7972 2^nd c2nd 7973 tpos ctpos 8211 WUnicwun 10697 sSet csts 17105 ndxcnx 17135 Basecbs 17153 Hom chom 17217 compcco 17218 Catccat 17617 oppCatcoppc 17664 CatCatccatc 18060

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-omul 8472 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-wun 10699 df-ni 10869 df-pli 10870 df-mi 10871 df-lti 10872 df-plpq 10905 df-mpq 10906 df-ltpq 10907 df-enq 10908 df-nq 10909 df-erq 10910 df-plq 10911 df-mq 10912 df-1nq 10913 df-rq 10914 df-ltnq 10915 df-np 10978 df-plp 10980 df-ltp 10982 df-enr 11052 df-nr 11053 df-c 11118 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-hom 17230 df-cco 17231 df-cat 17621 df-cid 17622 df-oppc 17665 df-catc 18061

This theorem is referenced by: (None)

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