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Theorem catcoppcclOLD 18070
Description: Obsolete proof of catcoppccl 18069 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 2724 . . . . . 6 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
3 eqid 2724 . . . . . 6 (Hom β€˜π‘‹) = (Hom β€˜π‘‹)
4 eqid 2724 . . . . . 6 (compβ€˜π‘‹) = (compβ€˜π‘‹)
5 catcoppccl.o . . . . . 6 𝑂 = (oppCatβ€˜π‘‹)
62, 3, 4, 5oppcval 17656 . . . . 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 18053 . . . . . . . 8 (πœ‘ β†’ 𝐡 = (π‘ˆ ∩ Cat))
121, 11eleqtrd 2827 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ (π‘ˆ ∩ Cat))
1312elin1d 4190 . . . . . 6 (πœ‘ β†’ 𝑋 ∈ π‘ˆ)
14 df-hom 17220 . . . . . . . 8 Hom = Slot 14
15 catcoppccl.2 . . . . . . . . 9 (πœ‘ β†’ Ο‰ ∈ π‘ˆ)
168, 15wunndx 17127 . . . . . . . 8 (πœ‘ β†’ ndx ∈ π‘ˆ)
1714, 8, 16wunstr 17120 . . . . . . 7 (πœ‘ β†’ (Hom β€˜ndx) ∈ π‘ˆ)
1814, 8, 13wunstr 17120 . . . . . . . 8 (πœ‘ β†’ (Hom β€˜π‘‹) ∈ π‘ˆ)
198, 18wuntpos 10725 . . . . . . 7 (πœ‘ β†’ tpos (Hom β€˜π‘‹) ∈ π‘ˆ)
208, 17, 19wunop 10713 . . . . . 6 (πœ‘ β†’ ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘‹)⟩ ∈ π‘ˆ)
218, 13, 20wunsets 17109 . . . . 5 (πœ‘ β†’ (𝑋 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘‹)⟩) ∈ π‘ˆ)
22 df-cco 17221 . . . . . . 7 comp = Slot 15
2322, 8, 16wunstr 17120 . . . . . 6 (πœ‘ β†’ (compβ€˜ndx) ∈ π‘ˆ)
24 df-base 17144 . . . . . . . . . 10 Base = Slot 1
2524, 8, 13wunstr 17120 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜π‘‹) ∈ π‘ˆ)
268, 25, 25wunxp 10715 . . . . . . . 8 (πœ‘ β†’ ((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)) ∈ π‘ˆ)
278, 26, 25wunxp 10715 . . . . . . 7 (πœ‘ β†’ (((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)) Γ— (Baseβ€˜π‘‹)) ∈ π‘ˆ)
2822, 8, 13wunstr 17120 . . . . . . . . . . . . . 14 (πœ‘ β†’ (compβ€˜π‘‹) ∈ π‘ˆ)
298, 28wunrn 10720 . . . . . . . . . . . . 13 (πœ‘ β†’ ran (compβ€˜π‘‹) ∈ π‘ˆ)
308, 29wununi 10697 . . . . . . . . . . . 12 (πœ‘ β†’ βˆͺ ran (compβ€˜π‘‹) ∈ π‘ˆ)
318, 30wundm 10719 . . . . . . . . . . 11 (πœ‘ β†’ dom βˆͺ ran (compβ€˜π‘‹) ∈ π‘ˆ)
328, 31wuncnv 10721 . . . . . . . . . 10 (πœ‘ β†’ β—‘dom βˆͺ ran (compβ€˜π‘‹) ∈ π‘ˆ)
338wun0 10709 . . . . . . . . . . 11 (πœ‘ β†’ βˆ… ∈ π‘ˆ)
348, 33wunsn 10707 . . . . . . . . . 10 (πœ‘ β†’ {βˆ…} ∈ π‘ˆ)
358, 32, 34wunun 10701 . . . . . . . . 9 (πœ‘ β†’ (β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) ∈ π‘ˆ)
368, 30wunrn 10720 . . . . . . . . 9 (πœ‘ β†’ ran βˆͺ ran (compβ€˜π‘‹) ∈ π‘ˆ)
378, 35, 36wunxp 10715 . . . . . . . 8 (πœ‘ β†’ ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)) ∈ π‘ˆ)
388, 37wunpw 10698 . . . . . . 7 (πœ‘ β†’ 𝒫 ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)) ∈ π‘ˆ)
39 tposssxp 8210 . . . . . . . . . . . 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 7437 . . . . . . . . . . . . . . 15 (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† βˆͺ ran (compβ€˜π‘‹)
41 dmss 5892 . . . . . . . . . . . . . . 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 5862 . . . . . . . . . . . . . 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 4171 . . . . . . . . . . . . . 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 5929 . . . . . . . . . . . . 13 ran (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† ran βˆͺ ran (compβ€˜π‘‹)
47 xpss12 5681 . . . . . . . . . . . . 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 3983 . . . . . . . . . . 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 5334 . . . . . . . . . . . 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 3142 . . . . . . . . 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 3142 . . . . . . . 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 2724 . . . . . . . . 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 8047 . . . . . . . 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 10718 . . . . . 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 10713 . . . . 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 17109 . . . 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 2825 . . 3 (πœ‘ β†’ 𝑂 ∈ π‘ˆ)
6212elin2d 4191 . . . 4 (πœ‘ β†’ 𝑋 ∈ Cat)
635oppccat 17667 . . . 4 (𝑋 ∈ Cat β†’ 𝑂 ∈ Cat)
6462, 63syl 17 . . 3 (πœ‘ β†’ 𝑂 ∈ Cat)
6561, 64elind 4186 . 2 (πœ‘ β†’ 𝑂 ∈ (π‘ˆ ∩ Cat))
6665, 11eleqtrrd 2828 1 (πœ‘ β†’ 𝑂 ∈ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   βˆͺ cun 3938   ∩ cin 3939   βŠ† wss 3940  βˆ…c0 4314  π’« cpw 4594  {csn 4620  βŸ¨cop 4626  βˆͺ cuni 4899   Γ— cxp 5664  β—‘ccnv 5665  dom cdm 5666  ran crn 5667  βŸΆwf 6529  β€˜cfv 6533  (class class class)co 7401   ∈ cmpo 7403  Ο‰com 7848  1st c1st 7966  2nd c2nd 7967  tpos ctpos 8205  WUnicwun 10691  1c1 11107  4c4 12266  5c5 12267  cdc 12674   sSet csts 17095  ndxcnx 17125  Basecbs 17143  Hom chom 17207  compcco 17208  Catccat 17607  oppCatcoppc 17654  CatCatccatc 18050
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-tpos 8206  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-omul 8466  df-er 8699  df-ec 8701  df-qs 8705  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-wun 10693  df-ni 10863  df-pli 10864  df-mi 10865  df-lti 10866  df-plpq 10899  df-mpq 10900  df-ltpq 10901  df-enq 10902  df-nq 10903  df-erq 10904  df-plq 10905  df-mq 10906  df-1nq 10907  df-rq 10908  df-ltnq 10909  df-np 10972  df-plp 10974  df-ltp 10976  df-enr 11046  df-nr 11047  df-c 11112  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-hom 17220  df-cco 17221  df-cat 17611  df-cid 17612  df-oppc 17655  df-catc 18051
This theorem is referenced by: (None)
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