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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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