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Theorem cnextf 24074
Description: Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
Hypotheses
Ref Expression
cnextf.1 𝐶 = 𝐽
cnextf.2 𝐵 = 𝐾
cnextf.3 (𝜑𝐽 ∈ Top)
cnextf.4 (𝜑𝐾 ∈ Haus)
cnextf.5 (𝜑𝐹:𝐴𝐵)
cnextf.a (𝜑𝐴𝐶)
cnextf.6 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)
cnextf.7 ((𝜑𝑥𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)
Assertion
Ref Expression
cnextf (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝜑,𝑥

Proof of Theorem cnextf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cnextf.3 . . . 4 (𝜑𝐽 ∈ Top)
2 cnextf.4 . . . 4 (𝜑𝐾 ∈ Haus)
3 cnextf.5 . . . 4 (𝜑𝐹:𝐴𝐵)
4 cnextf.a . . . 4 (𝜑𝐴𝐶)
5 cnextf.1 . . . . 5 𝐶 = 𝐽
6 cnextf.2 . . . . 5 𝐵 = 𝐾
75, 6cnextfun 24072 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹))
81, 2, 3, 4, 7syl22anc 839 . . 3 (𝜑 → Fun ((𝐽CnExt𝐾)‘𝐹))
9 dfdm3 5898 . . . 4 dom ((𝐽CnExt𝐾)‘𝐹) = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)}
10 simpl 482 . . . . . . 7 ((𝜑𝑥𝐶) → 𝜑)
11 cnextf.6 . . . . . . . . 9 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)
1211eleq2d 2827 . . . . . . . 8 (𝜑 → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥𝐶))
1312biimpar 477 . . . . . . 7 ((𝜑𝑥𝐶) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
14 cnextf.7 . . . . . . . 8 ((𝜑𝑥𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)
15 n0 4353 . . . . . . . 8 (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
1614, 15sylib 218 . . . . . . 7 ((𝜑𝑥𝐶) → ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
17 haustop 23339 . . . . . . . . . . . . . 14 (𝐾 ∈ Haus → 𝐾 ∈ Top)
182, 17syl 17 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ Top)
195, 6cnextfval 24070 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → ((𝐽CnExt𝐾)‘𝐹) = 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
201, 18, 3, 4, 19syl22anc 839 . . . . . . . . . . . 12 (𝜑 → ((𝐽CnExt𝐾)‘𝐹) = 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
2120eleq2d 2827 . . . . . . . . . . 11 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
22 opeliunxp 5752 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
2321, 22bitrdi 287 . . . . . . . . . 10 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
2423exbidv 1921 . . . . . . . . 9 (𝜑 → (∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ ∃𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
25 19.42v 1953 . . . . . . . . 9 (∃𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
2624, 25bitrdi 287 . . . . . . . 8 (𝜑 → (∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
2726biimpar 477 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) → ∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹))
2810, 13, 16, 27syl12anc 837 . . . . . 6 ((𝜑𝑥𝐶) → ∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹))
2926simprbda 498 . . . . . . 7 ((𝜑 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
3012adantr 480 . . . . . . 7 ((𝜑 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥𝐶))
3129, 30mpbid 232 . . . . . 6 ((𝜑 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)) → 𝑥𝐶)
3228, 31impbida 801 . . . . 5 (𝜑 → (𝑥𝐶 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)))
3332eqabdv 2875 . . . 4 (𝜑𝐶 = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)})
349, 33eqtr4id 2796 . . 3 (𝜑 → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶)
35 df-fn 6564 . . 3 (((𝐽CnExt𝐾)‘𝐹) Fn 𝐶 ↔ (Fun ((𝐽CnExt𝐾)‘𝐹) ∧ dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶))
368, 34, 35sylanbrc 583 . 2 (𝜑 → ((𝐽CnExt𝐾)‘𝐹) Fn 𝐶)
3720rneqd 5949 . . 3 (𝜑 → ran ((𝐽CnExt𝐾)‘𝐹) = ran 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
38 rniun 6167 . . . 4 ran 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) = 𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
39 vex 3484 . . . . . . . . 9 𝑥 ∈ V
4039snnz 4776 . . . . . . . 8 {𝑥} ≠ ∅
41 rnxp 6190 . . . . . . . 8 ({𝑥} ≠ ∅ → ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
4240, 41ax-mp 5 . . . . . . 7 ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)
4312biimpa 476 . . . . . . . 8 ((𝜑𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝑥𝐶)
446toptopon 22923 . . . . . . . . . . 11 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
4518, 44sylib 218 . . . . . . . . . 10 (𝜑𝐾 ∈ (TopOn‘𝐵))
4645adantr 480 . . . . . . . . 9 ((𝜑𝑥𝐶) → 𝐾 ∈ (TopOn‘𝐵))
475toptopon 22923 . . . . . . . . . . . 12 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
481, 47sylib 218 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝐶))
4948adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐶) → 𝐽 ∈ (TopOn‘𝐶))
504adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐶) → 𝐴𝐶)
51 simpr 484 . . . . . . . . . 10 ((𝜑𝑥𝐶) → 𝑥𝐶)
52 trnei 23900 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶𝑥𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
5352biimpa 476 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶𝑥𝐶) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
5449, 50, 51, 13, 53syl31anc 1375 . . . . . . . . 9 ((𝜑𝑥𝐶) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
553adantr 480 . . . . . . . . 9 ((𝜑𝑥𝐶) → 𝐹:𝐴𝐵)
56 flfelbas 24002 . . . . . . . . . . 11 (((𝐾 ∈ (TopOn‘𝐵) ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴𝐵) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) → 𝑦𝐵)
5756ex 412 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝐵) ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴𝐵) → (𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) → 𝑦𝐵))
5857ssrdv 3989 . . . . . . . . 9 ((𝐾 ∈ (TopOn‘𝐵) ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴𝐵) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ⊆ 𝐵)
5946, 54, 55, 58syl3anc 1373 . . . . . . . 8 ((𝜑𝑥𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ⊆ 𝐵)
6043, 59syldan 591 . . . . . . 7 ((𝜑𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ⊆ 𝐵)
6142, 60eqsstrid 4022 . . . . . 6 ((𝜑𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
6261ralrimiva 3146 . . . . 5 (𝜑 → ∀𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
63 iunss 5045 . . . . 5 ( 𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵 ↔ ∀𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
6462, 63sylibr 234 . . . 4 (𝜑 𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
6538, 64eqsstrid 4022 . . 3 (𝜑 → ran 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
6637, 65eqsstrd 4018 . 2 (𝜑 → ran ((𝐽CnExt𝐾)‘𝐹) ⊆ 𝐵)
67 df-f 6565 . 2 (((𝐽CnExt𝐾)‘𝐹):𝐶𝐵 ↔ (((𝐽CnExt𝐾)‘𝐹) Fn 𝐶 ∧ ran ((𝐽CnExt𝐾)‘𝐹) ⊆ 𝐵))
6836, 66, 67sylanbrc 583 1 (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  wral 3061  wss 3951  c0 4333  {csn 4626  cop 4632   cuni 4907   ciun 4991   × cxp 5683  dom cdm 5685  ran crn 5686  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  t crest 17465  Topctop 22899  TopOnctopon 22916  clsccl 23026  neicnei 23105  Hauscha 23316  Filcfil 23853   fLimf cflf 23943  CnExtccnext 24067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-pm 8869  df-rest 17467  df-fbas 21361  df-fg 21362  df-top 22900  df-topon 22917  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-haus 23323  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-cnext 24068
This theorem is referenced by:  cnextcn  24075  cnextfres1  24076
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