catcoppcclOLD - Metamath Proof Explorer

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Theorem catcoppcclOLD 18101

Description: Obsolete proof of catcoppccl 18100 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 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 17687	. . . . 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 18084	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
12	1, 11	eleqtrd 2827	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
13	12	elin1d 4193	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
14		df-hom 17251	. . . . . . . 8 ⊢ Hom = Slot ;14
15		catcoppccl.2	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ 𝑈)
16	8, 15	wunndx 17158	. . . . . . . 8 ⊢ (𝜑 → ndx ∈ 𝑈)
17	14, 8, 16	wunstr 17151	. . . . . . 7 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
18	14, 8, 13	wunstr 17151	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
19	8, 18	wuntpos 10752	. . . . . . 7 ⊢ (𝜑 → tpos (Hom ‘𝑋) ∈ 𝑈)
20	8, 17, 19	wunop 10740	. . . . . 6 ⊢ (𝜑 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩ ∈ 𝑈)
21	8, 13, 20	wunsets 17140	. . . . 5 ⊢ (𝜑 → (𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) ∈ 𝑈)
22		df-cco 17252	. . . . . . 7 ⊢ comp = Slot ;15
23	22, 8, 16	wunstr 17151	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
24		df-base 17175	. . . . . . . . . 10 ⊢ Base = Slot 1
25	24, 8, 13	wunstr 17151	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
26	8, 25, 25	wunxp 10742	. . . . . . . 8 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑋)) ∈ 𝑈)
27	8, 26, 25	wunxp 10742	. . . . . . 7 ⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋)) ∈ 𝑈)
28	22, 8, 13	wunstr 17151	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
29	8, 28	wunrn 10747	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
30	8, 29	wununi 10724	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈)
31	8, 30	wundm 10746	. . . . . . . . . . 11 ⊢ (𝜑 → dom ∪ ran (comp‘𝑋) ∈ 𝑈)
32	8, 31	wuncnv 10748	. . . . . . . . . 10 ⊢ (𝜑 → ^◡dom ∪ ran (comp‘𝑋) ∈ 𝑈)
33	8	wun0 10736	. . . . . . . . . . 11 ⊢ (𝜑 → ∅ ∈ 𝑈)
34	8, 33	wunsn 10734	. . . . . . . . . 10 ⊢ (𝜑 → {∅} ∈ 𝑈)
35	8, 32, 34	wunun 10728	. . . . . . . . 9 ⊢ (𝜑 → (^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) ∈ 𝑈)
36	8, 30	wunrn 10747	. . . . . . . . 9 ⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈)
37	8, 35, 36	wunxp 10742	. . . . . . . 8 ⊢ (𝜑 → ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
38	8, 37	wunpw 10725	. . . . . . 7 ⊢ (𝜑 → 𝒫 ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
39		tposssxp 8229	. . . . . . . . . . . 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 7449	. . . . . . . . . . . . . . 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 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) ∪ {∅}))
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 5688	. . . . . . . . . . . . 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 3983	. . . . . . . . . . 11 ⊢ tpos (⟨𝑦, (2^nd ‘𝑥)⟩(comp‘𝑋)(1^st ‘𝑥)) ⊆ ((^◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))
50		elpw2g 5342	. . . . . . . . . . . 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 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)))
54	53	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)))
55		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)))
56	55	fmpo 8066	. . . . . . . 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 10745	. . . . . 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 10740	. . . . 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 17140	. . . 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 2825	. . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑈)
62	12	elin2d 4194	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Cat)
63	5	oppccat 17698	. . . 4 ⊢ (𝑋 ∈ Cat → 𝑂 ∈ Cat)
64	62, 63	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
65	61, 64	elind 4189	. 2 ⊢ (𝜑 → 𝑂 ∈ (𝑈 ∩ Cat))
66	65, 11	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 3939 ∩ cin 3940 ⊆ wss 3941 ∅c0 4319 𝒫 cpw 4599 {csn 4625 ⟨cop 4631 ∪ cuni 4904 × cxp 5671 ^◡ccnv 5672 dom cdm 5673 ran crn 5674 ⟶wf 6539 ‘cfv 6543 (class class class)co 7413 ∈ cmpo 7415 ωcom 7865 1^st c1st 7985 2^nd c2nd 7986 tpos ctpos 8224 WUnicwun 10718 1c1 11134 4c4 12294 5c5 12295 cdc 12702 sSet csts 17126 ndxcnx 17156 Basecbs 17174 Hom chom 17238 compcco 17239 Catccat 17638 oppCatcoppc 17685 CatCatccatc 18081

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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-omul 8485 df-er 8718 df-ec 8720 df-qs 8724 df-map 8840 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-wun 10720 df-ni 10890 df-pli 10891 df-mi 10892 df-lti 10893 df-plpq 10926 df-mpq 10927 df-ltpq 10928 df-enq 10929 df-nq 10930 df-erq 10931 df-plq 10932 df-mq 10933 df-1nq 10934 df-rq 10935 df-ltnq 10936 df-np 10999 df-plp 11001 df-ltp 11003 df-enr 11073 df-nr 11074 df-c 11139 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-hom 17251 df-cco 17252 df-cat 17642 df-cid 17643 df-oppc 17686 df-catc 18082

This theorem is referenced by: (None)

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