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Theorem catcoppccl 16959
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 2760 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
3 eqid 2760 . . . . . 6 (Hom ‘𝑋) = (Hom ‘𝑋)
4 eqid 2760 . . . . . 6 (comp‘𝑋) = (comp‘𝑋)
5 catcoppccl.o . . . . . 6 𝑂 = (oppCat‘𝑋)
62, 3, 4, 5oppcval 16574 . . . . 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 inss1 3976 . . . . . . 7 (𝑈 ∩ Cat) ⊆ 𝑈
10 catcoppccl.c . . . . . . . . 9 𝐶 = (CatCat‘𝑈)
11 catcoppccl.b . . . . . . . . 9 𝐵 = (Base‘𝐶)
1210, 11, 8catcbas 16948 . . . . . . . 8 (𝜑𝐵 = (𝑈 ∩ Cat))
131, 12eleqtrd 2841 . . . . . . 7 (𝜑𝑋 ∈ (𝑈 ∩ Cat))
149, 13sseldi 3742 . . . . . 6 (𝜑𝑋𝑈)
15 df-hom 16168 . . . . . . . 8 Hom = Slot 14
16 catcoppccl.2 . . . . . . . . 9 (𝜑 → ω ∈ 𝑈)
178, 16wunndx 16080 . . . . . . . 8 (𝜑 → ndx ∈ 𝑈)
1815, 8, 17wunstr 16083 . . . . . . 7 (𝜑 → (Hom ‘ndx) ∈ 𝑈)
1915, 8, 14wunstr 16083 . . . . . . . 8 (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
208, 19wuntpos 9748 . . . . . . 7 (𝜑 → tpos (Hom ‘𝑋) ∈ 𝑈)
218, 18, 20wunop 9736 . . . . . 6 (𝜑 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩ ∈ 𝑈)
228, 14, 21wunsets 16102 . . . . 5 (𝜑 → (𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) ∈ 𝑈)
23 df-cco 16169 . . . . . . 7 comp = Slot 15
2423, 8, 17wunstr 16083 . . . . . 6 (𝜑 → (comp‘ndx) ∈ 𝑈)
25 df-base 16065 . . . . . . . . . 10 Base = Slot 1
2625, 8, 14wunstr 16083 . . . . . . . . 9 (𝜑 → (Base‘𝑋) ∈ 𝑈)
278, 26, 26wunxp 9738 . . . . . . . 8 (𝜑 → ((Base‘𝑋) × (Base‘𝑋)) ∈ 𝑈)
288, 27, 26wunxp 9738 . . . . . . 7 (𝜑 → (((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋)) ∈ 𝑈)
2923, 8, 14wunstr 16083 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑋) ∈ 𝑈)
308, 29wunrn 9743 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
318, 30wununi 9720 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑋) ∈ 𝑈)
328, 31wundm 9742 . . . . . . . . . . 11 (𝜑 → dom ran (comp‘𝑋) ∈ 𝑈)
338, 32wuncnv 9744 . . . . . . . . . 10 (𝜑dom ran (comp‘𝑋) ∈ 𝑈)
348wun0 9732 . . . . . . . . . . 11 (𝜑 → ∅ ∈ 𝑈)
358, 34wunsn 9730 . . . . . . . . . 10 (𝜑 → {∅} ∈ 𝑈)
368, 33, 35wunun 9724 . . . . . . . . 9 (𝜑 → (dom ran (comp‘𝑋) ∪ {∅}) ∈ 𝑈)
378, 31wunrn 9743 . . . . . . . . 9 (𝜑 → ran ran (comp‘𝑋) ∈ 𝑈)
388, 36, 37wunxp 9738 . . . . . . . 8 (𝜑 → ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)) ∈ 𝑈)
398, 38wunpw 9721 . . . . . . 7 (𝜑 → 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)) ∈ 𝑈)
40 tposssxp 7525 . . . . . . . . . . . 12 tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ((dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∪ {∅}) × ran (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)))
41 ovssunirn 6844 . . . . . . . . . . . . . . 15 (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ran (comp‘𝑋)
42 dmss 5478 . . . . . . . . . . . . . . 15 ((⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ran (comp‘𝑋) → dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ dom ran (comp‘𝑋))
4341, 42ax-mp 5 . . . . . . . . . . . . . 14 dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ dom ran (comp‘𝑋)
44 cnvss 5450 . . . . . . . . . . . . . 14 (dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ dom ran (comp‘𝑋) → dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ dom ran (comp‘𝑋))
45 unss1 3925 . . . . . . . . . . . . . 14 (dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ dom ran (comp‘𝑋) → (dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∪ {∅}) ⊆ (dom ran (comp‘𝑋) ∪ {∅}))
4643, 44, 45mp2b 10 . . . . . . . . . . . . 13 (dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∪ {∅}) ⊆ (dom ran (comp‘𝑋) ∪ {∅})
47 rnss 5509 . . . . . . . . . . . . . 14 ((⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ran (comp‘𝑋) → ran (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ran ran (comp‘𝑋))
4841, 47ax-mp 5 . . . . . . . . . . . . 13 ran (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ran ran (comp‘𝑋)
49 xpss12 5281 . . . . . . . . . . . . 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)))
5046, 48, 49mp2an 710 . . . . . . . . . . . 12 ((dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∪ {∅}) × ran (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥))) ⊆ ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋))
5140, 50sstri 3753 . . . . . . . . . . 11 tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋))
52 elpw2g 4976 . . . . . . . . . . . 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))))
5338, 52syl 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))))
5451, 53mpbiri 248 . . . . . . . . . 10 (𝜑 → tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∈ 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)))
5554ralrimivw 3105 . . . . . . . . 9 (𝜑 → ∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∈ 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)))
5655ralrimivw 3105 . . . . . . . 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)))
57 eqid 2760 . . . . . . . . 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)))
5857fmpt2 7405 . . . . . . . 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)))
5956, 58sylib 208 . . . . . . 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)))
608, 28, 39, 59wunf 9741 . . . . . 6 (𝜑 → (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥))) ∈ 𝑈)
618, 24, 60wunop 9736 . . . . 5 (𝜑 → ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)))⟩ ∈ 𝑈)
628, 22, 61wunsets 16102 . . . 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)
637, 62eqeltrd 2839 . . 3 (𝜑𝑂𝑈)
64 inss2 3977 . . . . 5 (𝑈 ∩ Cat) ⊆ Cat
6564, 13sseldi 3742 . . . 4 (𝜑𝑋 ∈ Cat)
665oppccat 16583 . . . 4 (𝑋 ∈ Cat → 𝑂 ∈ Cat)
6765, 66syl 17 . . 3 (𝜑𝑂 ∈ Cat)
6863, 67elind 3941 . 2 (𝜑𝑂 ∈ (𝑈 ∩ Cat))
6968, 12eleqtrrd 2842 1 (𝜑𝑂𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  wral 3050  cun 3713  cin 3714  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321  cop 4327   cuni 4588   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267  wf 6045  cfv 6049  (class class class)co 6813  cmpt2 6815  ωcom 7230  1st c1st 7331  2nd c2nd 7332  tpos ctpos 7520  WUnicwun 9714  1c1 10129  4c4 11264  5c5 11265  cdc 11685  ndxcnx 16056   sSet csts 16057  Basecbs 16059  Hom chom 16154  compcco 16155  Catccat 16526  oppCatcoppc 16572  CatCatccatc 16945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-tpos 7521  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-omul 7734  df-er 7911  df-ec 7913  df-qs 7917  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-wun 9716  df-ni 9886  df-pli 9887  df-mi 9888  df-lti 9889  df-plpq 9922  df-mpq 9923  df-ltpq 9924  df-enq 9925  df-nq 9926  df-erq 9927  df-plq 9928  df-mq 9929  df-1nq 9930  df-rq 9931  df-ltnq 9932  df-np 9995  df-plp 9997  df-ltp 9999  df-enr 10069  df-nr 10070  df-c 10134  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-hom 16168  df-cco 16169  df-cat 16530  df-cid 16531  df-oppc 16573  df-catc 16946
This theorem is referenced by: (None)
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