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Theorem catcoppccl 17360
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 2825 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
3 eqid 2825 . . . . . 6 (Hom ‘𝑋) = (Hom ‘𝑋)
4 eqid 2825 . . . . . 6 (comp‘𝑋) = (comp‘𝑋)
5 catcoppccl.o . . . . . 6 𝑂 = (oppCat‘𝑋)
62, 3, 4, 5oppcval 16975 . . . . 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 17349 . . . . . . . 8 (𝜑𝐵 = (𝑈 ∩ Cat))
121, 11eleqtrd 2919 . . . . . . 7 (𝜑𝑋 ∈ (𝑈 ∩ Cat))
1312elin1d 4178 . . . . . 6 (𝜑𝑋𝑈)
14 df-hom 16581 . . . . . . . 8 Hom = Slot 14
15 catcoppccl.2 . . . . . . . . 9 (𝜑 → ω ∈ 𝑈)
168, 15wunndx 16496 . . . . . . . 8 (𝜑 → ndx ∈ 𝑈)
1714, 8, 16wunstr 16499 . . . . . . 7 (𝜑 → (Hom ‘ndx) ∈ 𝑈)
1814, 8, 13wunstr 16499 . . . . . . . 8 (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
198, 18wuntpos 10148 . . . . . . 7 (𝜑 → tpos (Hom ‘𝑋) ∈ 𝑈)
208, 17, 19wunop 10136 . . . . . 6 (𝜑 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩ ∈ 𝑈)
218, 13, 20wunsets 16516 . . . . 5 (𝜑 → (𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) ∈ 𝑈)
22 df-cco 16582 . . . . . . 7 comp = Slot 15
2322, 8, 16wunstr 16499 . . . . . 6 (𝜑 → (comp‘ndx) ∈ 𝑈)
24 df-base 16481 . . . . . . . . . 10 Base = Slot 1
2524, 8, 13wunstr 16499 . . . . . . . . 9 (𝜑 → (Base‘𝑋) ∈ 𝑈)
268, 25, 25wunxp 10138 . . . . . . . 8 (𝜑 → ((Base‘𝑋) × (Base‘𝑋)) ∈ 𝑈)
278, 26, 25wunxp 10138 . . . . . . 7 (𝜑 → (((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋)) ∈ 𝑈)
2822, 8, 13wunstr 16499 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑋) ∈ 𝑈)
298, 28wunrn 10143 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
308, 29wununi 10120 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑋) ∈ 𝑈)
318, 30wundm 10142 . . . . . . . . . . 11 (𝜑 → dom ran (comp‘𝑋) ∈ 𝑈)
328, 31wuncnv 10144 . . . . . . . . . 10 (𝜑dom ran (comp‘𝑋) ∈ 𝑈)
338wun0 10132 . . . . . . . . . . 11 (𝜑 → ∅ ∈ 𝑈)
348, 33wunsn 10130 . . . . . . . . . 10 (𝜑 → {∅} ∈ 𝑈)
358, 32, 34wunun 10124 . . . . . . . . 9 (𝜑 → (dom ran (comp‘𝑋) ∪ {∅}) ∈ 𝑈)
368, 30wunrn 10143 . . . . . . . . 9 (𝜑 → ran ran (comp‘𝑋) ∈ 𝑈)
378, 35, 36wunxp 10138 . . . . . . . 8 (𝜑 → ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)) ∈ 𝑈)
388, 37wunpw 10121 . . . . . . 7 (𝜑 → 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)) ∈ 𝑈)
39 tposssxp 7890 . . . . . . . . . . . 12 tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ((dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∪ {∅}) × ran (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)))
40 ovssunirn 7187 . . . . . . . . . . . . . . 15 (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ran (comp‘𝑋)
41 dmss 5769 . . . . . . . . . . . . . . 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 5741 . . . . . . . . . . . . . 14 (dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ dom ran (comp‘𝑋) → dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ dom ran (comp‘𝑋))
44 unss1 4158 . . . . . . . . . . . . . 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 5808 . . . . . . . . . . . . 13 ran (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ran ran (comp‘𝑋)
47 xpss12 5568 . . . . . . . . . . . . 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 3979 . . . . . . . . . . 11 tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋))
50 elpw2g 5243 . . . . . . . . . . . 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 3187 . . . . . . . . 9 (𝜑 → ∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∈ 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)))
5453ralrimivw 3187 . . . . . . . 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 2825 . . . . . . . . 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 7760 . . . . . . . 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 10141 . . . . . 6 (𝜑 → (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥))) ∈ 𝑈)
598, 23, 58wunop 10136 . . . . 5 (𝜑 → ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)))⟩ ∈ 𝑈)
608, 21, 59wunsets 16516 . . . 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 2917 . . 3 (𝜑𝑂𝑈)
6212elin2d 4179 . . . 4 (𝜑𝑋 ∈ Cat)
635oppccat 16984 . . . 4 (𝑋 ∈ Cat → 𝑂 ∈ Cat)
6462, 63syl 17 . . 3 (𝜑𝑂 ∈ Cat)
6561, 64elind 4174 . 2 (𝜑𝑂 ∈ (𝑈 ∩ Cat))
6665, 11eleqtrrd 2920 1 (𝜑𝑂𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wcel 2107  wral 3142  cun 3937  cin 3938  wss 3939  c0 4294  𝒫 cpw 4541  {csn 4563  cop 4569   cuni 4836   × cxp 5551  ccnv 5552  dom cdm 5553  ran crn 5554  wf 6347  cfv 6351  (class class class)co 7151  cmpo 7153  ωcom 7571  1st c1st 7681  2nd c2nd 7682  tpos ctpos 7885  WUnicwun 10114  1c1 10530  4c4 11686  5c5 11687  cdc 12090  ndxcnx 16472   sSet csts 16473  Basecbs 16475  Hom chom 16568  compcco 16569  Catccat 16927  oppCatcoppc 16973  CatCatccatc 17346
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-tpos 7886  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-omul 8101  df-er 8282  df-ec 8284  df-qs 8288  df-map 8401  df-pm 8402  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-wun 10116  df-ni 10286  df-pli 10287  df-mi 10288  df-lti 10289  df-plpq 10322  df-mpq 10323  df-ltpq 10324  df-enq 10325  df-nq 10326  df-erq 10327  df-plq 10328  df-mq 10329  df-1nq 10330  df-rq 10331  df-ltnq 10332  df-np 10395  df-plp 10397  df-ltp 10399  df-enr 10469  df-nr 10470  df-c 10535  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12886  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-hom 16581  df-cco 16582  df-cat 16931  df-cid 16932  df-oppc 16974  df-catc 17347
This theorem is referenced by: (None)
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