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Theorem catcoppcclOLD 18098
Description: Obsolete proof of catcoppccl 18097 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 2727 . . . . . 6 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
3 eqid 2727 . . . . . 6 (Hom β€˜π‘‹) = (Hom β€˜π‘‹)
4 eqid 2727 . . . . . 6 (compβ€˜π‘‹) = (compβ€˜π‘‹)
5 catcoppccl.o . . . . . 6 𝑂 = (oppCatβ€˜π‘‹)
62, 3, 4, 5oppcval 17684 . . . . 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 18081 . . . . . . . 8 (πœ‘ β†’ 𝐡 = (π‘ˆ ∩ Cat))
121, 11eleqtrd 2830 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ (π‘ˆ ∩ Cat))
1312elin1d 4194 . . . . . 6 (πœ‘ β†’ 𝑋 ∈ π‘ˆ)
14 df-hom 17248 . . . . . . . 8 Hom = Slot 14
15 catcoppccl.2 . . . . . . . . 9 (πœ‘ β†’ Ο‰ ∈ π‘ˆ)
168, 15wunndx 17155 . . . . . . . 8 (πœ‘ β†’ ndx ∈ π‘ˆ)
1714, 8, 16wunstr 17148 . . . . . . 7 (πœ‘ β†’ (Hom β€˜ndx) ∈ π‘ˆ)
1814, 8, 13wunstr 17148 . . . . . . . 8 (πœ‘ β†’ (Hom β€˜π‘‹) ∈ π‘ˆ)
198, 18wuntpos 10749 . . . . . . 7 (πœ‘ β†’ tpos (Hom β€˜π‘‹) ∈ π‘ˆ)
208, 17, 19wunop 10737 . . . . . 6 (πœ‘ β†’ ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘‹)⟩ ∈ π‘ˆ)
218, 13, 20wunsets 17137 . . . . 5 (πœ‘ β†’ (𝑋 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘‹)⟩) ∈ π‘ˆ)
22 df-cco 17249 . . . . . . 7 comp = Slot 15
2322, 8, 16wunstr 17148 . . . . . 6 (πœ‘ β†’ (compβ€˜ndx) ∈ π‘ˆ)
24 df-base 17172 . . . . . . . . . 10 Base = Slot 1
2524, 8, 13wunstr 17148 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜π‘‹) ∈ π‘ˆ)
268, 25, 25wunxp 10739 . . . . . . . 8 (πœ‘ β†’ ((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)) ∈ π‘ˆ)
278, 26, 25wunxp 10739 . . . . . . 7 (πœ‘ β†’ (((Baseβ€˜π‘‹) Γ— (Baseβ€˜π‘‹)) Γ— (Baseβ€˜π‘‹)) ∈ π‘ˆ)
2822, 8, 13wunstr 17148 . . . . . . . . . . . . . 14 (πœ‘ β†’ (compβ€˜π‘‹) ∈ π‘ˆ)
298, 28wunrn 10744 . . . . . . . . . . . . 13 (πœ‘ β†’ ran (compβ€˜π‘‹) ∈ π‘ˆ)
308, 29wununi 10721 . . . . . . . . . . . 12 (πœ‘ β†’ βˆͺ ran (compβ€˜π‘‹) ∈ π‘ˆ)
318, 30wundm 10743 . . . . . . . . . . 11 (πœ‘ β†’ dom βˆͺ ran (compβ€˜π‘‹) ∈ π‘ˆ)
328, 31wuncnv 10745 . . . . . . . . . 10 (πœ‘ β†’ β—‘dom βˆͺ ran (compβ€˜π‘‹) ∈ π‘ˆ)
338wun0 10733 . . . . . . . . . . 11 (πœ‘ β†’ βˆ… ∈ π‘ˆ)
348, 33wunsn 10731 . . . . . . . . . 10 (πœ‘ β†’ {βˆ…} ∈ π‘ˆ)
358, 32, 34wunun 10725 . . . . . . . . 9 (πœ‘ β†’ (β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) ∈ π‘ˆ)
368, 30wunrn 10744 . . . . . . . . 9 (πœ‘ β†’ ran βˆͺ ran (compβ€˜π‘‹) ∈ π‘ˆ)
378, 35, 36wunxp 10739 . . . . . . . 8 (πœ‘ β†’ ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)) ∈ π‘ˆ)
388, 37wunpw 10722 . . . . . . 7 (πœ‘ β†’ 𝒫 ((β—‘dom βˆͺ ran (compβ€˜π‘‹) βˆͺ {βˆ…}) Γ— ran βˆͺ ran (compβ€˜π‘‹)) ∈ π‘ˆ)
39 tposssxp 8229 . . . . . . . . . . . 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 7450 . . . . . . . . . . . . . . 15 (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† βˆͺ ran (compβ€˜π‘‹)
41 dmss 5899 . . . . . . . . . . . . . . 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 5869 . . . . . . . . . . . . . 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 4175 . . . . . . . . . . . . . 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 5936 . . . . . . . . . . . . 13 ran (βŸ¨π‘¦, (2nd β€˜π‘₯)⟩(compβ€˜π‘‹)(1st β€˜π‘₯)) βŠ† ran βˆͺ ran (compβ€˜π‘‹)
47 xpss12 5687 . . . . . . . . . . . . 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 691 . . . . . . . . . . . 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 3987 . . . . . . . . . . 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 5340 . . . . . . . . . . . 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 3145 . . . . . . . . 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 3145 . . . . . . . 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 2727 . . . . . . . . 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 8066 . . . . . . . 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 10742 . . . . . 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 10737 . . . . 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 17137 . . . 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 2828 . . 3 (πœ‘ β†’ 𝑂 ∈ π‘ˆ)
6212elin2d 4195 . . . 4 (πœ‘ β†’ 𝑋 ∈ Cat)
635oppccat 17695 . . . 4 (𝑋 ∈ Cat β†’ 𝑂 ∈ Cat)
6462, 63syl 17 . . 3 (πœ‘ β†’ 𝑂 ∈ Cat)
6561, 64elind 4190 . 2 (πœ‘ β†’ 𝑂 ∈ (π‘ˆ ∩ Cat))
6665, 11eleqtrrd 2831 1 (πœ‘ β†’ 𝑂 ∈ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056   βˆͺ cun 3942   ∩ cin 3943   βŠ† wss 3944  βˆ…c0 4318  π’« cpw 4598  {csn 4624  βŸ¨cop 4630  βˆͺ cuni 4903   Γ— cxp 5670  β—‘ccnv 5671  dom cdm 5672  ran crn 5673  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  Ο‰com 7864  1st c1st 7985  2nd c2nd 7986  tpos ctpos 8224  WUnicwun 10715  1c1 11131  4c4 12291  5c5 12292  cdc 12699   sSet csts 17123  ndxcnx 17153  Basecbs 17171  Hom chom 17235  compcco 17236  Catccat 17635  oppCatcoppc 17682  CatCatccatc 18078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-omul 8485  df-er 8718  df-ec 8720  df-qs 8724  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-wun 10717  df-ni 10887  df-pli 10888  df-mi 10889  df-lti 10890  df-plpq 10923  df-mpq 10924  df-ltpq 10925  df-enq 10926  df-nq 10927  df-erq 10928  df-plq 10929  df-mq 10930  df-1nq 10931  df-rq 10932  df-ltnq 10933  df-np 10996  df-plp 10998  df-ltp 11000  df-enr 11070  df-nr 11071  df-c 11136  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-fz 13509  df-struct 17107  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-hom 17248  df-cco 17249  df-cat 17639  df-cid 17640  df-oppc 17683  df-catc 18079
This theorem is referenced by: (None)
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