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Theorem catcoppcclOLD 18078
Description: Obsolete proof of catcoppccl 18077 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 (𝜑𝑈 ∈ WUni)
catcoppccl.2 (𝜑 → ω ∈ 𝑈)
catcoppccl.3 (𝜑𝑋𝐵)
Assertion
Ref Expression
catcoppcclOLD (𝜑𝑂𝐵)

Proof of Theorem catcoppcclOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcoppccl.3 . . . . 5 (𝜑𝑋𝐵)
2 eqid 2731 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
3 eqid 2731 . . . . . 6 (Hom ‘𝑋) = (Hom ‘𝑋)
4 eqid 2731 . . . . . 6 (comp‘𝑋) = (comp‘𝑋)
5 catcoppccl.o . . . . . 6 𝑂 = (oppCat‘𝑋)
62, 3, 4, 5oppcval 17664 . . . . 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 18061 . . . . . . . 8 (𝜑𝐵 = (𝑈 ∩ Cat))
121, 11eleqtrd 2834 . . . . . . 7 (𝜑𝑋 ∈ (𝑈 ∩ Cat))
1312elin1d 4198 . . . . . 6 (𝜑𝑋𝑈)
14 df-hom 17228 . . . . . . . 8 Hom = Slot 14
15 catcoppccl.2 . . . . . . . . 9 (𝜑 → ω ∈ 𝑈)
168, 15wunndx 17135 . . . . . . . 8 (𝜑 → ndx ∈ 𝑈)
1714, 8, 16wunstr 17128 . . . . . . 7 (𝜑 → (Hom ‘ndx) ∈ 𝑈)
1814, 8, 13wunstr 17128 . . . . . . . 8 (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
198, 18wuntpos 10735 . . . . . . 7 (𝜑 → tpos (Hom ‘𝑋) ∈ 𝑈)
208, 17, 19wunop 10723 . . . . . 6 (𝜑 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩ ∈ 𝑈)
218, 13, 20wunsets 17117 . . . . 5 (𝜑 → (𝑋 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑋)⟩) ∈ 𝑈)
22 df-cco 17229 . . . . . . 7 comp = Slot 15
2322, 8, 16wunstr 17128 . . . . . 6 (𝜑 → (comp‘ndx) ∈ 𝑈)
24 df-base 17152 . . . . . . . . . 10 Base = Slot 1
2524, 8, 13wunstr 17128 . . . . . . . . 9 (𝜑 → (Base‘𝑋) ∈ 𝑈)
268, 25, 25wunxp 10725 . . . . . . . 8 (𝜑 → ((Base‘𝑋) × (Base‘𝑋)) ∈ 𝑈)
278, 26, 25wunxp 10725 . . . . . . 7 (𝜑 → (((Base‘𝑋) × (Base‘𝑋)) × (Base‘𝑋)) ∈ 𝑈)
2822, 8, 13wunstr 17128 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑋) ∈ 𝑈)
298, 28wunrn 10730 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
308, 29wununi 10707 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑋) ∈ 𝑈)
318, 30wundm 10729 . . . . . . . . . . 11 (𝜑 → dom ran (comp‘𝑋) ∈ 𝑈)
328, 31wuncnv 10731 . . . . . . . . . 10 (𝜑dom ran (comp‘𝑋) ∈ 𝑈)
338wun0 10719 . . . . . . . . . . 11 (𝜑 → ∅ ∈ 𝑈)
348, 33wunsn 10717 . . . . . . . . . 10 (𝜑 → {∅} ∈ 𝑈)
358, 32, 34wunun 10711 . . . . . . . . 9 (𝜑 → (dom ran (comp‘𝑋) ∪ {∅}) ∈ 𝑈)
368, 30wunrn 10730 . . . . . . . . 9 (𝜑 → ran ran (comp‘𝑋) ∈ 𝑈)
378, 35, 36wunxp 10725 . . . . . . . 8 (𝜑 → ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)) ∈ 𝑈)
388, 37wunpw 10708 . . . . . . 7 (𝜑 → 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)) ∈ 𝑈)
39 tposssxp 8221 . . . . . . . . . . . 12 tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ((dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∪ {∅}) × ran (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)))
40 ovssunirn 7448 . . . . . . . . . . . . . . 15 (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ran (comp‘𝑋)
41 dmss 5902 . . . . . . . . . . . . . . 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 5872 . . . . . . . . . . . . . 14 (dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ dom ran (comp‘𝑋) → dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ dom ran (comp‘𝑋))
44 unss1 4179 . . . . . . . . . . . . . 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 5939 . . . . . . . . . . . . 13 ran (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ran ran (comp‘𝑋)
47 xpss12 5691 . . . . . . . . . . . . 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 689 . . . . . . . . . . . 12 ((dom (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∪ {∅}) × ran (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥))) ⊆ ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋))
4939, 48sstri 3991 . . . . . . . . . . 11 tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ⊆ ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋))
50 elpw2g 5344 . . . . . . . . . . . 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 258 . . . . . . . . . 10 (𝜑 → tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∈ 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)))
5352ralrimivw 3149 . . . . . . . . 9 (𝜑 → ∀𝑦 ∈ (Base‘𝑋)tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)) ∈ 𝒫 ((dom ran (comp‘𝑋) ∪ {∅}) × ran ran (comp‘𝑋)))
5453ralrimivw 3149 . . . . . . . 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 2731 . . . . . . . . 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 8058 . . . . . . . 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 217 . . . . . . 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 10728 . . . . . 6 (𝜑 → (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥))) ∈ 𝑈)
598, 23, 58wunop 10723 . . . . 5 (𝜑 → ⟨(comp‘ndx), (𝑥 ∈ ((Base‘𝑋) × (Base‘𝑋)), 𝑦 ∈ (Base‘𝑋) ↦ tpos (⟨𝑦, (2nd𝑥)⟩(comp‘𝑋)(1st𝑥)))⟩ ∈ 𝑈)
608, 21, 59wunsets 17117 . . . 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 2832 . . 3 (𝜑𝑂𝑈)
6212elin2d 4199 . . . 4 (𝜑𝑋 ∈ Cat)
635oppccat 17675 . . . 4 (𝑋 ∈ Cat → 𝑂 ∈ Cat)
6462, 63syl 17 . . 3 (𝜑𝑂 ∈ Cat)
6561, 64elind 4194 . 2 (𝜑𝑂 ∈ (𝑈 ∩ Cat))
6665, 11eleqtrrd 2835 1 (𝜑𝑂𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  wcel 2105  wral 3060  cun 3946  cin 3947  wss 3948  c0 4322  𝒫 cpw 4602  {csn 4628  cop 4634   cuni 4908   × cxp 5674  ccnv 5675  dom cdm 5676  ran crn 5677  wf 6539  cfv 6543  (class class class)co 7412  cmpo 7414  ωcom 7859  1st c1st 7977  2nd c2nd 7978  tpos ctpos 8216  WUnicwun 10701  1c1 11117  4c4 12276  5c5 12277  cdc 12684   sSet csts 17103  ndxcnx 17133  Basecbs 17151  Hom chom 17215  compcco 17216  Catccat 17615  oppCatcoppc 17662  CatCatccatc 18058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-tpos 8217  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-oadd 8476  df-omul 8477  df-er 8709  df-ec 8711  df-qs 8715  df-map 8828  df-pm 8829  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-wun 10703  df-ni 10873  df-pli 10874  df-mi 10875  df-lti 10876  df-plpq 10909  df-mpq 10910  df-ltpq 10911  df-enq 10912  df-nq 10913  df-erq 10914  df-plq 10915  df-mq 10916  df-1nq 10917  df-rq 10918  df-ltnq 10919  df-np 10982  df-plp 10984  df-ltp 10986  df-enr 11056  df-nr 11057  df-c 11122  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-fz 13492  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-hom 17228  df-cco 17229  df-cat 17619  df-cid 17620  df-oppc 17663  df-catc 18059
This theorem is referenced by: (None)
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