Theorem catcoppcclOLD 18004

Description: Obsolete proof of catcoppccl 18003 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 2736	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
3		eqid 2736	. . . . . 6 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
4		eqid 2736	. . . . . 6 ⊢ (comp‘𝑋) = (comp‘𝑋)
5		catcoppccl.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝑋)
6	2, 3, 4, 5	oppcval 17593	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑂 = ((𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) sSet ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘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, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘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 17987	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
12	1, 11	eleqtrd 2840	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
13	12	elin1d 4158	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
14		df-hom 17157	. . . . . . . 8 ⊢ Hom = Slot ;14
15		catcoppccl.2	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ 𝑈)
16	8, 15	wunndx 17067	. . . . . . . 8 ⊢ (𝜑 → ndx ∈ 𝑈)
17	14, 8, 16	wunstr 17060	. . . . . . 7 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
18	14, 8, 13	wunstr 17060	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
19	8, 18	wuntpos 10670	. . . . . . 7 ⊢ (𝜑 → tpos (Hom ‘𝑋) ∈ 𝑈)
20	8, 17, 19	wunop 10658	. . . . . 6 ⊢ (𝜑 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩ ∈ 𝑈)
21	8, 13, 20	wunsets 17049	. . . . 5 ⊢ (𝜑 → (𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) ∈ 𝑈)
22		df-cco 17158	. . . . . . 7 ⊢ comp = Slot ;15
23	22, 8, 16	wunstr 17060	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
24		df-base 17084	. . . . . . . . . 10 ⊢ Base = Slot 1
25	24, 8, 13	wunstr 17060	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
26	8, 25, 25	wunxp 10660	. . . . . . . 8 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑋)) ∈ 𝑈)
27	8, 26, 25	wunxp 10660	. . . . . . 7 ⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋)) ∈ 𝑈)
28	22, 8, 13	wunstr 17060	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
29	8, 28	wunrn 10665	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
30	8, 29	wununi 10642	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈)
31	8, 30	wundm 10664	. . . . . . . . . . 11 ⊢ (𝜑 → dom ∪ ran (comp‘𝑋) ∈ 𝑈)
32	8, 31	wuncnv 10666	. . . . . . . . . 10 ⊢ (𝜑 → ◡dom ∪ ran (comp‘𝑋) ∈ 𝑈)
33	8	wun0 10654	. . . . . . . . . . 11 ⊢ (𝜑 → ∅ ∈ 𝑈)
34	8, 33	wunsn 10652	. . . . . . . . . 10 ⊢ (𝜑 → {∅} ∈ 𝑈)
35	8, 32, 34	wunun 10646	. . . . . . . . 9 ⊢ (𝜑 → (◡dom ∪ ran (comp‘𝑋) ∪ {∅}) ∈ 𝑈)
36	8, 30	wunrn 10665	. . . . . . . . 9 ⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈)
37	8, 35, 36	wunxp 10660	. . . . . . . 8 ⊢ (𝜑 → ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
38	8, 37	wunpw 10643	. . . . . . 7 ⊢ (𝜑 → 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈)
39		tposssxp 8161	. . . . . . . . . . . 12 ⊢ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ((◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∪ {∅}) × ran (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)))
40		ovssunirn 7393	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ∪ ran (comp‘𝑋)
41		dmss 5858	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ∪ ran (comp‘𝑋) → dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ dom ∪ ran (comp‘𝑋))
42	40, 41	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ dom ∪ ran (comp‘𝑋)
43		cnvss 5828	. . . . . . . . . . . . . 14 ⊢ (dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ dom ∪ ran (comp‘𝑋) → ◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ◡dom ∪ ran (comp‘𝑋))
44		unss1 4139	. . . . . . . . . . . . . 14 ⊢ (◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ◡dom ∪ ran (comp‘𝑋) → (◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∪ {∅}) ⊆ (◡dom ∪ ran (comp‘𝑋) ∪ {∅}))
45	42, 43, 44	mp2b 10	. . . . . . . . . . . . 13 ⊢ (◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∪ {∅}) ⊆ (◡dom ∪ ran (comp‘𝑋) ∪ {∅})
46	40	rnssi 5895	. . . . . . . . . . . . 13 ⊢ ran (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ran ∪ ran (comp‘𝑋)
47		xpss12 5648	. . . . . . . . . . . . 13 ⊢ (((◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∪ {∅}) ⊆ (◡dom ∪ ran (comp‘𝑋) ∪ {∅}) ∧ ran (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ran ∪ ran (comp‘𝑋)) → ((◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∪ {∅}) × ran (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥))) ⊆ ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
48	45, 46, 47	mp2an 690	. . . . . . . . . . . 12 ⊢ ((◡dom (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∪ {∅}) × ran (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥))) ⊆ ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))
49	39, 48	sstri 3953	. . . . . . . . . . 11 ⊢ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))
50		elpw2g 5301	. . . . . . . . . . . 12 ⊢ (((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ∈ 𝑈 → (tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∈ 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ↔ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))))
51	37, 50	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∈ 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ↔ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ⊆ ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋))))
52	49, 51	mpbiri 257	. . . . . . . . . 10 ⊢ (𝜑 → tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∈ 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
53	52	ralrimivw 3147	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∈ 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
54	53	ralrimivw 3147	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋))∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∈ 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
55		eqid 2736	. . . . . . . . 9 ⊢ (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥))) = (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)))
56	55	fmpo 8000	. . . . . . . 8 ⊢ (∀𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋))∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)) ∈ 𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)) ↔ (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘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, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥))):(((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋))⟶𝒫 ((◡dom ∪ ran (comp‘𝑋) ∪ {∅}) × ran ∪ ran (comp‘𝑋)))
58	8, 27, 38, 57	wunf 10663	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥))) ∈ 𝑈)
59	8, 23, 58	wunop 10658	. . . . 5 ⊢ (𝜑 → ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)))⟩ ∈ 𝑈)
60	8, 21, 59	wunsets 17049	. . . 4 ⊢ (𝜑 → ((𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) sSet ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd ‘𝑥)⟩(comp‘𝑋)(1st ‘𝑥)))⟩) ∈ 𝑈)
61	7, 60	eqeltrd 2838	. . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑈)
62	12	elin2d 4159	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Cat)
63	5	oppccat 17604	. . . 4 ⊢ (𝑋 ∈ Cat → 𝑂 ∈ Cat)
64	62, 63	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
65	61, 64	elind 4154	. 2 ⊢ (𝜑 → 𝑂 ∈ (𝑈 ∩ Cat))
66	65, 11	eleqtrrd 2841	1 ⊢ (𝜑 → 𝑂 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  ∀wral 3064   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  {csn 4586  ⟨cop 4592  ∪ cuni 4865   × cxp 5631  ^◡ccnv 5632  dom cdm 5633  ran crn 5634  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  ωcom 7802  1^st c1st 7919  2^nd c2nd 7920  tpos ctpos 8156  WUnicwun 10636  1c1 11052  4c4 12210  5c5 12211  ;cdc 12618   sSet csts 17035  ndxcnx 17065  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  oppCatcoppc 17591  CatCatccatc 17984

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-er 8648  df-ec 8650  df-qs 8654  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-wun 10638  df-ni 10808  df-pli 10809  df-mi 10810  df-lti 10811  df-plpq 10844  df-mpq 10845  df-ltpq 10846  df-enq 10847  df-nq 10848  df-erq 10849  df-plq 10850  df-mq 10851  df-1nq 10852  df-rq 10853  df-ltnq 10854  df-np 10917  df-plp 10919  df-ltp 10921  df-enr 10991  df-nr 10992  df-c 11057  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-cat 17548  df-cid 17549  df-oppc 17592  df-catc 17985

This theorem is referenced by: (None)