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Theorem catcoppccl 17356
Description: The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
catcoppccl.c 𝐶 = (CatCat‘𝑈)
catcoppccl.b 𝐵 = (Base‘𝐶)
catcoppccl.o 𝑂 = (oppCat‘𝑋)
catcoppccl.1 (𝜑𝑈 ∈ WUni)
catcoppccl.2 (𝜑 → ω ∈ 𝑈)
catcoppccl.3 (𝜑𝑋𝐵)
Assertion
Ref Expression
catcoppccl (𝜑𝑂𝐵)

Proof of Theorem catcoppccl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcoppccl.3 . . . . 5 (𝜑𝑋𝐵)
2 eqid 2818 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
3 eqid 2818 . . . . . 6 (Hom ‘𝑋) = (Hom ‘𝑋)
4 eqid 2818 . . . . . 6 (comp‘𝑋) = (comp‘𝑋)
5 catcoppccl.o . . . . . 6 𝑂 = (oppCat‘𝑋)
62, 3, 4, 5oppcval 16971 . . . . 5 (𝑋𝐵𝑂 = ((𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) sSet ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)))⟩))
71, 6syl 17 . . . 4 (𝜑𝑂 = ((𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) sSet ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)))⟩))
8 catcoppccl.1 . . . . 5 (𝜑𝑈 ∈ WUni)
9 catcoppccl.c . . . . . . . . 9 𝐶 = (CatCat‘𝑈)
10 catcoppccl.b . . . . . . . . 9 𝐵 = (Base‘𝐶)
119, 10, 8catcbas 17345 . . . . . . . 8 (𝜑𝐵 = (𝑈 ∩ Cat))
121, 11eleqtrd 2912 . . . . . . 7 (𝜑𝑋 ∈ (𝑈 ∩ Cat))
1312elin1d 4172 . . . . . 6 (𝜑𝑋𝑈)
14 df-hom 16577 . . . . . . . 8 Hom = Slot 14
15 catcoppccl.2 . . . . . . . . 9 (𝜑 → ω ∈ 𝑈)
168, 15wunndx 16492 . . . . . . . 8 (𝜑 → ndx ∈ 𝑈)
1714, 8, 16wunstr 16495 . . . . . . 7 (𝜑 → (Hom ‘ndx) ∈ 𝑈)
1814, 8, 13wunstr 16495 . . . . . . . 8 (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
198, 18wuntpos 10144 . . . . . . 7 (𝜑 → tpos (Hom ‘𝑋) ∈ 𝑈)
208, 17, 19wunop 10132 . . . . . 6 (𝜑 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩ ∈ 𝑈)
218, 13, 20wunsets 16512 . . . . 5 (𝜑 → (𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) ∈ 𝑈)
22 df-cco 16578 . . . . . . 7 comp = Slot 15
2322, 8, 16wunstr 16495 . . . . . 6 (𝜑 → (comp‘ndx) ∈ 𝑈)
24 df-base 16477 . . . . . . . . . 10 Base = Slot 1
2524, 8, 13wunstr 16495 . . . . . . . . 9 (𝜑 → (Base‘𝑋) ∈ 𝑈)
268, 25, 25wunxp 10134 . . . . . . . 8 (𝜑 → ((Base‘𝑋) × (Base‘𝑋)) ∈ 𝑈)
278, 26, 25wunxp 10134 . . . . . . 7 (𝜑 → (((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋)) ∈ 𝑈)
2822, 8, 13wunstr 16495 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑋) ∈ 𝑈)
298, 28wunrn 10139 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
308, 29wununi 10116 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑋) ∈ 𝑈)
318, 30wundm 10138 . . . . . . . . . . 11 (𝜑 → dom ran (comp‘𝑋) ∈ 𝑈)
328, 31wuncnv 10140 . . . . . . . . . 10 (𝜑dom ran (comp‘𝑋) ∈ 𝑈)
338wun0 10128 . . . . . . . . . . 11 (𝜑 → ∅ ∈ 𝑈)
348, 33wunsn 10126 . . . . . . . . . 10 (𝜑 → {∅} ∈ 𝑈)
358, 32, 34wunun 10120 . . . . . . . . 9 (𝜑 → (dom ran (comp‘𝑋) ∪ {∅}) ∈ 𝑈)
368, 30wunrn 10139 . . . . . . . . 9 (𝜑 → ran ran (comp‘𝑋) ∈ 𝑈)
378, 35, 36wunxp 10134 . . . . . . . 8 (𝜑 → ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)) ∈ 𝑈)
388, 37wunpw 10117 . . . . . . 7 (𝜑 → 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)) ∈ 𝑈)
39 tposssxp 7885 . . . . . . . . . . . 12 tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ((dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∪ {∅}) × ran (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)))
40 ovssunirn 7181 . . . . . . . . . . . . . . 15 (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ran (comp‘𝑋)
41 dmss 5764 . . . . . . . . . . . . . . 15 ((⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ran (comp‘𝑋) → dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ dom ran (comp‘𝑋))
4240, 41ax-mp 5 . . . . . . . . . . . . . 14 dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ dom ran (comp‘𝑋)
43 cnvss 5736 . . . . . . . . . . . . . 14 (dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ dom ran (comp‘𝑋) → dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ dom ran (comp‘𝑋))
44 unss1 4152 . . . . . . . . . . . . . 14 (dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ dom ran (comp‘𝑋) → (dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∪ {∅}) ⊆ (dom ran (comp‘𝑋) ∪ {∅}))
4542, 43, 44mp2b 10 . . . . . . . . . . . . 13 (dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∪ {∅}) ⊆ (dom ran (comp‘𝑋) ∪ {∅})
4640rnssi 5803 . . . . . . . . . . . . 13 ran (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ran ran (comp‘𝑋)
47 xpss12 5563 . . . . . . . . . . . . 13 (((dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∪ {∅}) ⊆ (dom ran (comp‘𝑋) ∪ {∅}) ∧ ran (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ran ran (comp‘𝑋)) → ((dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∪ {∅}) × ran (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥))) ⊆ ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)))
4845, 46, 47mp2an 688 . . . . . . . . . . . 12 ((dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∪ {∅}) × ran (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥))) ⊆ ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋))
4939, 48sstri 3973 . . . . . . . . . . 11 tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋))
50 elpw2g 5238 . . . . . . . . . . . 12 (((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)) ∈ 𝑈 → (tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∈ 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)) ↔ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋))))
5137, 50syl 17 . . . . . . . . . . 11 (𝜑 → (tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∈ 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)) ↔ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋))))
5249, 51mpbiri 259 . . . . . . . . . 10 (𝜑 → tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∈ 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)))
5352ralrimivw 3180 . . . . . . . . 9 (𝜑 → ∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∈ 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)))
5453ralrimivw 3180 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋))∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∈ 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)))
55 eqid 2818 . . . . . . . . 9 (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥))) = (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)))
5655fmpo 7755 . . . . . . . 8 (∀𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋))∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∈ 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)) ↔ (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥))):(((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋))⟶𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)))
5754, 56sylib 219 . . . . . . 7 (𝜑 → (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥))):(((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋))⟶𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)))
588, 27, 38, 57wunf 10137 . . . . . 6 (𝜑 → (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥))) ∈ 𝑈)
598, 23, 58wunop 10132 . . . . 5 (𝜑 → ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)))⟩ ∈ 𝑈)
608, 21, 59wunsets 16512 . . . 4 (𝜑 → ((𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) sSet ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)))⟩) ∈ 𝑈)
617, 60eqeltrd 2910 . . 3 (𝜑𝑂𝑈)
6212elin2d 4173 . . . 4 (𝜑𝑋 ∈ Cat)
635oppccat 16980 . . . 4 (𝑋 ∈ Cat → 𝑂 ∈ Cat)
6462, 63syl 17 . . 3 (𝜑𝑂 ∈ Cat)
6561, 64elind 4168 . 2 (𝜑𝑂 ∈ (𝑈 ∩ Cat))
6665, 11eleqtrrd 2913 1 (𝜑𝑂𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  wral 3135  cun 3931  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535  {csn 4557  cop 4563   cuni 4830   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  ωcom 7569  1st c1st 7676  2nd c2nd 7677  tpos ctpos 7880  WUnicwun 10110  1c1 10526  4c4 11682  5c5 11683  cdc 12086  ndxcnx 16468   sSet csts 16469  Basecbs 16471  Hom chom 16564  compcco 16565  Catccat 16923  oppCatcoppc 16969  CatCatccatc 17342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-tpos 7881  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-omul 8096  df-er 8278  df-ec 8280  df-qs 8284  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-wun 10112  df-ni 10282  df-pli 10283  df-mi 10284  df-lti 10285  df-plpq 10318  df-mpq 10319  df-ltpq 10320  df-enq 10321  df-nq 10322  df-erq 10323  df-plq 10324  df-mq 10325  df-1nq 10326  df-rq 10327  df-ltnq 10328  df-np 10391  df-plp 10393  df-ltp 10395  df-enr 10465  df-nr 10466  df-c 10531  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12881  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-hom 16577  df-cco 16578  df-cat 16927  df-cid 16928  df-oppc 16970  df-catc 17343
This theorem is referenced by: (None)
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