catcoppccl - Metamath Proof Explorer

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Theorem catcoppccl 18103

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 2725	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
3		eqid 2725	. . . . . 6 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
4		eqid 2725	. . . . . 6 ⊢ (comp‘𝑋) = (comp‘𝑋)
5		catcoppccl.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝑋)
6	2, 3, 4, 5	oppcval 17690	. . . . 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 18098	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
12		homid 17390	. . . . . . . 8 ⊢ Hom = Slot (Hom ‘ndx)
13		catcoppccl.2	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ 𝑈)
14	8, 13	wunndx 17161	. . . . . . . 8 ⊢ (𝜑 → ndx ∈ 𝑈)
15	12, 8, 14	wunstr 17154	. . . . . . 7 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
16	9, 10, 8, 1	catchomcl 18101	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
17	8, 16	wuntpos 10755	. . . . . . 7 ⊢ (𝜑 → tpos (Hom ‘𝑋) ∈ 𝑈)
18	8, 15, 17	wunop 10743	. . . . . 6 ⊢ (𝜑 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩ ∈ 𝑈)
19	8, 11, 18	wunsets 17143	. . . . 5 ⊢ (𝜑 → (𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) ∈ 𝑈)
20		ccoid 17392	. . . . . . 7 ⊢ comp = Slot (comp‘ndx)
21	20, 8, 14	wunstr 17154	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
22	9, 10, 8, 1	catcbaselcl 18100	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
23	8, 22, 22	wunxp 10745	. . . . . . . 8 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑋)) ∈ 𝑈)
24	8, 23, 22	wunxp 10745	. . . . . . 7 ⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋)) ∈ 𝑈)
25	9, 10, 8, 1	catcccocl 18102	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
26	8, 25	wunrn 10750	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
27	8, 26	wununi 10727	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈)
28	8, 27	wundm 10749	. . . . . . . . . . 11 ⊢ (𝜑 → dom ∪ ran (comp‘𝑋) ∈ 𝑈)
29	8, 28	wuncnv 10751	. . . . . . . . . 10 ⊢ (𝜑 → ^◡dom ∪ ran (comp‘𝑋) ∈ 𝑈)
30	8	wun0 10739	. . . . . . . . . . 11 ⊢ (𝜑 → ∅ ∈ 𝑈)
31	8, 30	wunsn 10737	. . . . . . . . . 10 ⊢ (𝜑 → {∅} ∈ 𝑈)
32	8, 29, 31	wunun 10731	. . . . . . . . 9 ⊢ (𝜑 → (^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) ∈ 𝑈)
33	8, 27	wunrn 10750	. . . . . . . . 9 ⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈)
34	8, 32, 33	wunxp 10745	. . . . . . . 8 ⊢ (𝜑 → ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
35	8, 34	wunpw 10728	. . . . . . 7 ⊢ (𝜑 → 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
36		tposssxp 8232	. . . . . . . . . . . 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 7451	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ∪ ran (comp‘𝑋)
38		dmss 5899	. . . . . . . . . . . . . . 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 5869	. . . . . . . . . . . . . 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 4173	. . . . . . . . . . . . . 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 5936	. . . . . . . . . . . . 13 ⊢ ran (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ran ∪ ran (comp‘𝑋)
44		xpss12 5687	. . . . . . . . . . . . 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 690	. . . . . . . . . . . 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 3982	. . . . . . . . . . 11 ⊢ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))
47		elpw2g 5341	. . . . . . . . . . . 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 257	. . . . . . . . . 10 ⊢ (𝜑 → tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∈ 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
50	49	ralrimivw 3140	. . . . . . . . 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 3140	. . . . . . . 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 2725	. . . . . . . . 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 8068	. . . . . . . 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 10748	. . . . . 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 10743	. . . . 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 17143	. . . 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 2825	. . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑈)
59	9, 10, 8	catcbas 18087	. . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
60	1, 59	eleqtrd 2827	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
61	60	elin2d 4193	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Cat)
62	5	oppccat 17701	. . . 4 ⊢ (𝑋 ∈ Cat → 𝑂 ∈ Cat)
63	61, 62	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
64	58, 63	elind 4188	. 2 ⊢ (𝜑 → 𝑂 ∈ (𝑈 ∩ Cat))
65	64, 59	eleqtrrd 2828	1 ⊢ (𝜑 → 𝑂 ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ∪ cun 3938 ∩ cin 3939 ⊆ wss 3940 ∅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 7415 ∈ cmpo 7417 ωcom 7867 1^st c1st 7987 2^nd c2nd 7988 tpos ctpos 8227 WUnicwun 10721 sSet csts 17129 ndxcnx 17159 Basecbs 17177 Hom chom 17241 compcco 17242 Catccat 17641 oppCatcoppc 17688 CatCatccatc 18084

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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-oadd 8487 df-omul 8488 df-er 8721 df-ec 8723 df-qs 8727 df-map 8843 df-pm 8844 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-wun 10723 df-ni 10893 df-pli 10894 df-mi 10895 df-lti 10896 df-plpq 10929 df-mpq 10930 df-ltpq 10931 df-enq 10932 df-nq 10933 df-erq 10934 df-plq 10935 df-mq 10936 df-1nq 10937 df-rq 10938 df-ltnq 10939 df-np 11002 df-plp 11004 df-ltp 11006 df-enr 11076 df-nr 11077 df-c 11142 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-hom 17254 df-cco 17255 df-cat 17645 df-cid 17646 df-oppc 17689 df-catc 18085

This theorem is referenced by: (None)

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