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Theorem catcoppcclOLD 18073
Description: Obsolete proof of catcoppccl 18072 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 𝐢 = (CatCatβ€˜π‘ˆ)
catcoppccl.b 𝐡 = (Baseβ€˜πΆ)
catcoppccl.o 𝑂 = (oppCatβ€˜π‘‹)
catcoppccl.1 (πœ‘ β†’ π‘ˆ ∈ WUni)
catcoppccl.2 (πœ‘ β†’ Ο‰ ∈ π‘ˆ)
catcoppccl.3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
Assertion
Ref Expression
catcoppcclOLD (πœ‘ β†’ 𝑂 ∈ 𝐡)

Proof of Theorem catcoppcclOLD
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcoppccl.3 . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝐡)
2 eqid 2731 . . . . . 6 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
3 eqid 2731 . . . . . 6 (Hom β€˜π‘‹) = (Hom β€˜π‘‹)
4 eqid 2731 . . . . . 6 (compβ€˜π‘‹) = (compβ€˜π‘‹)
5 catcoppccl.o . . . . . 6 𝑂 = (oppCatβ€˜π‘‹)
62, 3, 4, 5oppcval 17662 . . . . 5 (𝑋 ∈ 𝐡 β†’ 𝑂 = ((𝑋 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘‹)⟩) sSet ⟨(compβ€˜ndx), (π‘₯ ∈ ((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)), 𝑦 ∈ (Baseβ€˜π‘‹) ↦ tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)))⟩))
71, 6syl 17 . . . 4 (πœ‘ β†’ 𝑂 = ((𝑋 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘‹)⟩) sSet ⟨(compβ€˜ndx), (π‘₯ ∈ ((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)), 𝑦 ∈ (Baseβ€˜π‘‹) ↦ tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)))⟩))
8 catcoppccl.1 . . . . 5 (πœ‘ β†’ π‘ˆ ∈ WUni)
9 catcoppccl.c . . . . . . . . 9 𝐢 = (CatCatβ€˜π‘ˆ)
10 catcoppccl.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΆ)
119, 10, 8catcbas 18056 . . . . . . . 8 (πœ‘ β†’ 𝐡 = (π‘ˆ ∩ Cat))
121, 11eleqtrd 2834 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ (π‘ˆ ∩ Cat))
1312elin1d 4198 . . . . . 6 (πœ‘ β†’ 𝑋 ∈ π‘ˆ)
14 df-hom 17226 . . . . . . . 8 Hom = Slot 14
15 catcoppccl.2 . . . . . . . . 9 (πœ‘ β†’ Ο‰ ∈ π‘ˆ)
168, 15wunndx 17133 . . . . . . . 8 (πœ‘ β†’ ndx ∈ π‘ˆ)
1714, 8, 16wunstr 17126 . . . . . . 7 (πœ‘ β†’ (Hom β€˜ndx) ∈ π‘ˆ)
1814, 8, 13wunstr 17126 . . . . . . . 8 (πœ‘ β†’ (Hom β€˜π‘‹) ∈ π‘ˆ)
198, 18wuntpos 10733 . . . . . . 7 (πœ‘ β†’ tpos (Hom β€˜π‘‹) ∈ π‘ˆ)
208, 17, 19wunop 10721 . . . . . 6 (πœ‘ β†’ ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘‹)⟩ ∈ π‘ˆ)
218, 13, 20wunsets 17115 . . . . 5 (πœ‘ β†’ (𝑋 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘‹)⟩) ∈ π‘ˆ)
22 df-cco 17227 . . . . . . 7 comp = Slot 15
2322, 8, 16wunstr 17126 . . . . . 6 (πœ‘ β†’ (compβ€˜ndx) ∈ π‘ˆ)
24 df-base 17150 . . . . . . . . . 10 Base = Slot 1
2524, 8, 13wunstr 17126 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜π‘‹) ∈ π‘ˆ)
268, 25, 25wunxp 10723 . . . . . . . 8 (πœ‘ β†’ ((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)) ∈ π‘ˆ)
278, 26, 25wunxp 10723 . . . . . . 7 (πœ‘ β†’ (((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)) Γ— (Baseβ€˜π‘‹)) ∈ π‘ˆ)
2822, 8, 13wunstr 17126 . . . . . . . . . . . . . 14 (πœ‘ β†’ (compβ€˜π‘‹) ∈ π‘ˆ)
298, 28wunrn 10728 . . . . . . . . . . . . 13 (πœ‘ β†’ ran (compβ€˜π‘‹) ∈ π‘ˆ)
308, 29wununi 10705 . . . . . . . . . . . 12 (πœ‘ β†’ βˆͺ ran (compβ€˜π‘‹) ∈ π‘ˆ)
318, 30wundm 10727 . . . . . . . . . . 11 (πœ‘ β†’ dom βˆͺ ran (compβ€˜π‘‹) ∈ π‘ˆ)
328, 31wuncnv 10729 . . . . . . . . . 10 (πœ‘ β†’ β—‘dom βˆͺ ran (compβ€˜π‘‹) ∈ π‘ˆ)
338wun0 10717 . . . . . . . . . . 11 (πœ‘ β†’ βˆ… ∈ π‘ˆ)
348, 33wunsn 10715 . . . . . . . . . 10 (πœ‘ β†’ {βˆ…} ∈ π‘ˆ)
358, 32, 34wunun 10709 . . . . . . . . 9 (πœ‘ β†’ (β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) ∈ π‘ˆ)
368, 30wunrn 10728 . . . . . . . . 9 (πœ‘ β†’ ran βˆͺ ran (compβ€˜π‘‹) ∈ π‘ˆ)
378, 35, 36wunxp 10723 . . . . . . . 8 (πœ‘ β†’ ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)) ∈ π‘ˆ)
388, 37wunpw 10706 . . . . . . 7 (πœ‘ β†’ 𝒫 ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)) ∈ π‘ˆ)
39 tposssxp 8219 . . . . . . . . . . . 12 tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† ((β—‘dom (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βˆͺ {βˆ…}) Γ— ran (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)))
40 ovssunirn 7448 . . . . . . . . . . . . . . 15 (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† βˆͺ ran (compβ€˜π‘‹)
41 dmss 5902 . . . . . . . . . . . . . . 15 ((βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† βˆͺ ran (compβ€˜π‘‹) β†’ dom (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† dom βˆͺ ran (compβ€˜π‘‹))
4240, 41ax-mp 5 . . . . . . . . . . . . . 14 dom (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† dom βˆͺ ran (compβ€˜π‘‹)
43 cnvss 5872 . . . . . . . . . . . . . 14 (dom (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† dom βˆͺ ran (compβ€˜π‘‹) β†’ β—‘dom (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† β—‘dom βˆͺ ran (compβ€˜π‘‹))
44 unss1 4179 . . . . . . . . . . . . . 14 (β—‘dom (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† β—‘dom βˆͺ ran (compβ€˜π‘‹) β†’ (β—‘dom (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βˆͺ {βˆ…}) βŠ† (β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}))
4542, 43, 44mp2b 10 . . . . . . . . . . . . 13 (β—‘dom (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βˆͺ {βˆ…}) βŠ† (β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…})
4640rnssi 5939 . . . . . . . . . . . . 13 ran (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† ran βˆͺ ran (compβ€˜π‘‹)
47 xpss12 5691 . . . . . . . . . . . . 13 (((β—‘dom (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βˆͺ {βˆ…}) βŠ† (β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) ∧ ran (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† ran βˆͺ ran (compβ€˜π‘‹)) β†’ ((β—‘dom (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βˆͺ {βˆ…}) Γ— ran (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯))) βŠ† ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)))
4845, 46, 47mp2an 689 . . . . . . . . . . . 12 ((β—‘dom (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βˆͺ {βˆ…}) Γ— ran (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯))) βŠ† ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹))
4939, 48sstri 3991 . . . . . . . . . . 11 tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹))
50 elpw2g 5344 . . . . . . . . . . . 12 (((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)) ∈ π‘ˆ β†’ (tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) ∈ 𝒫 ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)) ↔ tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹))))
5137, 50syl 17 . . . . . . . . . . 11 (πœ‘ β†’ (tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) ∈ 𝒫 ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)) ↔ tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹))))
5249, 51mpbiri 258 . . . . . . . . . 10 (πœ‘ β†’ tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) ∈ 𝒫 ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)))
5352ralrimivw 3149 . . . . . . . . 9 (πœ‘ β†’ βˆ€π‘¦ ∈ (Baseβ€˜π‘‹)tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) ∈ 𝒫 ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)))
5453ralrimivw 3149 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘₯ ∈ ((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹))βˆ€π‘¦ ∈ (Baseβ€˜π‘‹)tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) ∈ 𝒫 ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)))
55 eqid 2731 . . . . . . . . 9 (π‘₯ ∈ ((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)), 𝑦 ∈ (Baseβ€˜π‘‹) ↦ tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯))) = (π‘₯ ∈ ((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)), 𝑦 ∈ (Baseβ€˜π‘‹) ↦ tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)))
5655fmpo 8058 . . . . . . . 8 (βˆ€π‘₯ ∈ ((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹))βˆ€π‘¦ ∈ (Baseβ€˜π‘‹)tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) ∈ 𝒫 ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)) ↔ (π‘₯ ∈ ((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)), 𝑦 ∈ (Baseβ€˜π‘‹) ↦ tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯))):(((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)) Γ— (Baseβ€˜π‘‹))βŸΆπ’« ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)))
5754, 56sylib 217 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ ((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)), 𝑦 ∈ (Baseβ€˜π‘‹) ↦ tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯))):(((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)) Γ— (Baseβ€˜π‘‹))βŸΆπ’« ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)))
588, 27, 38, 57wunf 10726 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ ((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)), 𝑦 ∈ (Baseβ€˜π‘‹) ↦ tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯))) ∈ π‘ˆ)
598, 23, 58wunop 10721 . . . . 5 (πœ‘ β†’ ⟨(compβ€˜ndx), (π‘₯ ∈ ((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)), 𝑦 ∈ (Baseβ€˜π‘‹) ↦ tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)))⟩ ∈ π‘ˆ)
608, 21, 59wunsets 17115 . . . 4 (πœ‘ β†’ ((𝑋 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘‹)⟩) sSet ⟨(compβ€˜ndx), (π‘₯ ∈ ((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)), 𝑦 ∈ (Baseβ€˜π‘‹) ↦ tpos (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)))⟩) ∈ π‘ˆ)
617, 60eqeltrd 2832 . . 3 (πœ‘ β†’ 𝑂 ∈ π‘ˆ)
6212elin2d 4199 . . . 4 (πœ‘ β†’ 𝑋 ∈ Cat)
635oppccat 17673 . . . 4 (𝑋 ∈ Cat β†’ 𝑂 ∈ Cat)
6462, 63syl 17 . . 3 (πœ‘ β†’ 𝑂 ∈ Cat)
6561, 64elind 4194 . 2 (πœ‘ β†’ 𝑂 ∈ (π‘ˆ ∩ Cat))
6665, 11eleqtrrd 2835 1 (πœ‘ β†’ 𝑂 ∈ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  {csn 4628  βŸ¨cop 4634  βˆͺ cuni 4908   Γ— cxp 5674  β—‘ccnv 5675  dom cdm 5676  ran crn 5677  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412   ∈ cmpo 7414  Ο‰com 7859  1st c1st 7977  2nd c2nd 7978  tpos ctpos 8214  WUnicwun 10699  1c1 11115  4c4 12274  5c5 12275  cdc 12682   sSet csts 17101  ndxcnx 17131  Basecbs 17149  Hom chom 17213  compcco 17214  Catccat 17613  oppCatcoppc 17660  CatCatccatc 18053
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 9640  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
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 8215  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-oadd 8474  df-omul 8475  df-er 8707  df-ec 8709  df-qs 8713  df-map 8826  df-pm 8827  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-wun 10701  df-ni 10871  df-pli 10872  df-mi 10873  df-lti 10874  df-plpq 10907  df-mpq 10908  df-ltpq 10909  df-enq 10910  df-nq 10911  df-erq 10912  df-plq 10913  df-mq 10914  df-1nq 10915  df-rq 10916  df-ltnq 10917  df-np 10980  df-plp 10982  df-ltp 10984  df-enr 11054  df-nr 11055  df-c 11120  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-uz 12828  df-fz 13490  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-hom 17226  df-cco 17227  df-cat 17617  df-cid 17618  df-oppc 17661  df-catc 18054
This theorem is referenced by: (None)
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