catcoppcclOLD - Metamath Proof Explorer

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Theorem catcoppcclOLD 18073

Description: Obsolete proof of catcoppccl 18072 as of 13-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

**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
catcoppcclOLD	⊢ (𝜑 → 𝑂 ∈ 𝐵)

Proof of Theorem catcoppcclOLD 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 2731	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
3		eqid 2731	. . . . . 6 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
4		eqid 2731	. . . . . 6 ⊢ (comp‘𝑋) = (comp‘𝑋)
5		catcoppccl.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝑋)
6	2, 3, 4, 5	oppcval 17662	. . . . 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	. . . . . . . . 9 ⊢ 𝐶 = (CatCat‘𝑈)
10		catcoppccl.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
11	9, 10, 8	catcbas 18056	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
12	1, 11	eleqtrd 2834	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
13	12	elin1d 4198	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
14		df-hom 17226	. . . . . . . 8 ⊢ Hom = Slot ;14
15		catcoppccl.2	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ 𝑈)
16	8, 15	wunndx 17133	. . . . . . . 8 ⊢ (𝜑 → ndx ∈ 𝑈)
17	14, 8, 16	wunstr 17126	. . . . . . 7 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
18	14, 8, 13	wunstr 17126	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
19	8, 18	wuntpos 10733	. . . . . . 7 ⊢ (𝜑 → tpos (Hom ‘𝑋) ∈ 𝑈)
20	8, 17, 19	wunop 10721	. . . . . 6 ⊢ (𝜑 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩ ∈ 𝑈)
21	8, 13, 20	wunsets 17115	. . . . 5 ⊢ (𝜑 → (𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) ∈ 𝑈)
22		df-cco 17227	. . . . . . 7 ⊢ comp = Slot ;15
23	22, 8, 16	wunstr 17126	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
24		df-base 17150	. . . . . . . . . 10 ⊢ Base = Slot 1
25	24, 8, 13	wunstr 17126	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
26	8, 25, 25	wunxp 10723	. . . . . . . 8 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑋)) ∈ 𝑈)
27	8, 26, 25	wunxp 10723	. . . . . . 7 ⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋)) ∈ 𝑈)
28	22, 8, 13	wunstr 17126	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
29	8, 28	wunrn 10728	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
30	8, 29	wununi 10705	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈)
31	8, 30	wundm 10727	. . . . . . . . . . 11 ⊢ (𝜑 → dom ∪ ran (comp‘𝑋) ∈ 𝑈)
32	8, 31	wuncnv 10729	. . . . . . . . . 10 ⊢ (𝜑 → ^◡dom ∪ ran (comp‘𝑋) ∈ 𝑈)
33	8	wun0 10717	. . . . . . . . . . 11 ⊢ (𝜑 → ∅ ∈ 𝑈)
34	8, 33	wunsn 10715	. . . . . . . . . 10 ⊢ (𝜑 → {∅} ∈ 𝑈)
35	8, 32, 34	wunun 10709	. . . . . . . . 9 ⊢ (𝜑 → (^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) ∈ 𝑈)
36	8, 30	wunrn 10728	. . . . . . . . 9 ⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈)
37	8, 35, 36	wunxp 10723	. . . . . . . 8 ⊢ (𝜑 → ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
38	8, 37	wunpw 10706	. . . . . . 7 ⊢ (𝜑 → 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
39		tposssxp 8219	. . . . . . . . . . . 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)))
40		ovssunirn 7448	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ∪ ran (comp‘𝑋)
41		dmss 5902	. . . . . . . . . . . . . . 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))
42	40, 41	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ dom ∪ ran (comp‘𝑋)
43		cnvss 5872	. . . . . . . . . . . . . 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))
44		unss1 4179	. . . . . . . . . . . . . 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) ∪ {∅}))
45	42, 43, 44	mp2b 10	. . . . . . . . . . . . 13 ⊢ (^◡dom (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∪ {∅}) ⊆ (^◡dom ∪ ran (comp‘𝑋) ∪ {∅})
46	40	rnssi 5939	. . . . . . . . . . . . 13 ⊢ ran (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ran ∪ ran (comp‘𝑋)
47		xpss12 5691	. . . . . . . . . . . . 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)))
48	45, 46, 47	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))
49	39, 48	sstri 3991	. . . . . . . . . . 11 ⊢ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))
50		elpw2g 5344	. . . . . . . . . . . 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))))
51	37, 50	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))))
52	49, 51	mpbiri 258	. . . . . . . . . 10 ⊢ (𝜑 → tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∈ 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
53	52	ralrimivw 3149	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∈ 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
54	53	ralrimivw 3149	. . . . . . . 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)))
55		eqid 2731	. . . . . . . . 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)))
56	55	fmpo 8058	. . . . . . . 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)))
57	54, 56	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)))
58	8, 27, 38, 57	wunf 10726	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥))) ∈ 𝑈)
59	8, 23, 58	wunop 10721	. . . . 5 ⊢ (𝜑 → ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)))⟩ ∈ 𝑈)
60	8, 21, 59	wunsets 17115	. . . 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)
61	7, 60	eqeltrd 2832	. . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑈)
62	12	elin2d 4199	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Cat)
63	5	oppccat 17673	. . . 4 ⊢ (𝑋 ∈ Cat → 𝑂 ∈ Cat)
64	62, 63	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
65	61, 64	elind 4194	. 2 ⊢ (𝜑 → 𝑂 ∈ (𝑈 ∩ Cat))
66	65, 11	eleqtrrd 2835	1 ⊢ (𝜑 → 𝑂 ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {csn 4628 ⟨cop 4634 ∪ cuni 4908 × cxp 5674 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 ωcom 7859 1^st c1st 7977 2^nd c2nd 7978 tpos ctpos 8214 WUnicwun 10699 1c1 11115 4c4 12274 5c5 12275 cdc 12682 sSet csts 17101 ndxcnx 17131 Basecbs 17149 Hom chom 17213 compcco 17214 Catccat 17613 oppCatcoppc 17660 CatCatccatc 18053

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-oadd 8474 df-omul 8475 df-er 8707 df-ec 8709 df-qs 8713 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-wun 10701 df-ni 10871 df-pli 10872 df-mi 10873 df-lti 10874 df-plpq 10907 df-mpq 10908 df-ltpq 10909 df-enq 10910 df-nq 10911 df-erq 10912 df-plq 10913 df-mq 10914 df-1nq 10915 df-rq 10916 df-ltnq 10917 df-np 10980 df-plp 10982 df-ltp 10984 df-enr 11054 df-nr 11055 df-c 11120 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-hom 17226 df-cco 17227 df-cat 17617 df-cid 17618 df-oppc 17661 df-catc 18054

This theorem is referenced by: (None)

W3C validator