catcoppcclOLD - Metamath Proof Explorer

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Theorem catcoppcclOLD 18064

Description: Obsolete proof of catcoppccl 18063 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 2732	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
3		eqid 2732	. . . . . 6 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
4		eqid 2732	. . . . . 6 ⊢ (comp‘𝑋) = (comp‘𝑋)
5		catcoppccl.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝑋)
6	2, 3, 4, 5	oppcval 17653	. . . . 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 18047	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
12	1, 11	eleqtrd 2835	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
13	12	elin1d 4197	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
14		df-hom 17217	. . . . . . . 8 ⊢ Hom = Slot ;14
15		catcoppccl.2	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ 𝑈)
16	8, 15	wunndx 17124	. . . . . . . 8 ⊢ (𝜑 → ndx ∈ 𝑈)
17	14, 8, 16	wunstr 17117	. . . . . . 7 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
18	14, 8, 13	wunstr 17117	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
19	8, 18	wuntpos 10725	. . . . . . 7 ⊢ (𝜑 → tpos (Hom ‘𝑋) ∈ 𝑈)
20	8, 17, 19	wunop 10713	. . . . . 6 ⊢ (𝜑 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩ ∈ 𝑈)
21	8, 13, 20	wunsets 17106	. . . . 5 ⊢ (𝜑 → (𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) ∈ 𝑈)
22		df-cco 17218	. . . . . . 7 ⊢ comp = Slot ;15
23	22, 8, 16	wunstr 17117	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
24		df-base 17141	. . . . . . . . . 10 ⊢ Base = Slot 1
25	24, 8, 13	wunstr 17117	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
26	8, 25, 25	wunxp 10715	. . . . . . . 8 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑋)) ∈ 𝑈)
27	8, 26, 25	wunxp 10715	. . . . . . 7 ⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋)) ∈ 𝑈)
28	22, 8, 13	wunstr 17117	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
29	8, 28	wunrn 10720	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
30	8, 29	wununi 10697	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈)
31	8, 30	wundm 10719	. . . . . . . . . . 11 ⊢ (𝜑 → dom ∪ ran (comp‘𝑋) ∈ 𝑈)
32	8, 31	wuncnv 10721	. . . . . . . . . 10 ⊢ (𝜑 → ^◡dom ∪ ran (comp‘𝑋) ∈ 𝑈)
33	8	wun0 10709	. . . . . . . . . . 11 ⊢ (𝜑 → ∅ ∈ 𝑈)
34	8, 33	wunsn 10707	. . . . . . . . . 10 ⊢ (𝜑 → {∅} ∈ 𝑈)
35	8, 32, 34	wunun 10701	. . . . . . . . 9 ⊢ (𝜑 → (^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) ∈ 𝑈)
36	8, 30	wunrn 10720	. . . . . . . . 9 ⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈)
37	8, 35, 36	wunxp 10715	. . . . . . . 8 ⊢ (𝜑 → ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
38	8, 37	wunpw 10698	. . . . . . 7 ⊢ (𝜑 → 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
39		tposssxp 8211	. . . . . . . . . . . 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 7441	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ∪ ran (comp‘𝑋)
41		dmss 5900	. . . . . . . . . . . . . . 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 5870	. . . . . . . . . . . . . 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 4178	. . . . . . . . . . . . . 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 5937	. . . . . . . . . . . . 13 ⊢ ran (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ran ∪ ran (comp‘𝑋)
47		xpss12 5690	. . . . . . . . . . . . 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 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))
49	39, 48	sstri 3990	. . . . . . . . . . 11 ⊢ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))
50		elpw2g 5343	. . . . . . . . . . . 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 257	. . . . . . . . . 10 ⊢ (𝜑 → tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ∈ 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
53	52	ralrimivw 3150	. . . . . . . . 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 3150	. . . . . . . 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 2732	. . . . . . . . 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 8050	. . . . . . . 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 10718	. . . . . 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 10713	. . . . 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 17106	. . . 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 2833	. . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑈)
62	12	elin2d 4198	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Cat)
63	5	oppccat 17664	. . . 4 ⊢ (𝑋 ∈ Cat → 𝑂 ∈ Cat)
64	62, 63	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
65	61, 64	elind 4193	. 2 ⊢ (𝜑 → 𝑂 ∈ (𝑈 ∩ Cat))
66	65, 11	eleqtrrd 2836	1 ⊢ (𝜑 → 𝑂 ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 {csn 4627 ⟨cop 4633 ∪ cuni 4907 × cxp 5673 ^◡ccnv 5674 dom cdm 5675 ran crn 5676 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 ωcom 7851 1^st c1st 7969 2^nd c2nd 7970 tpos ctpos 8206 WUnicwun 10691 1c1 11107 4c4 12265 5c5 12266 cdc 12673 sSet csts 17092 ndxcnx 17122 Basecbs 17140 Hom chom 17204 compcco 17205 Catccat 17604 oppCatcoppc 17651 CatCatccatc 18044

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-omul 8467 df-er 8699 df-ec 8701 df-qs 8705 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-wun 10693 df-ni 10863 df-pli 10864 df-mi 10865 df-lti 10866 df-plpq 10899 df-mpq 10900 df-ltpq 10901 df-enq 10902 df-nq 10903 df-erq 10904 df-plq 10905 df-mq 10906 df-1nq 10907 df-rq 10908 df-ltnq 10909 df-np 10972 df-plp 10974 df-ltp 10976 df-enr 11046 df-nr 11047 df-c 11112 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-hom 17217 df-cco 17218 df-cat 17608 df-cid 17609 df-oppc 17652 df-catc 18045

This theorem is referenced by: (None)

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