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Theorem cnextf 22963
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 22961 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹))
81, 2, 3, 4, 7syl22anc 839 . . 3 (𝜑 → Fun ((𝐽CnExt𝐾)‘𝐹))
9 dfdm3 5756 . . . 4 dom ((𝐽CnExt𝐾)‘𝐹) = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)}
10 simpl 486 . . . . . . 7 ((𝜑𝑥𝐶) → 𝜑)
11 cnextf.6 . . . . . . . . 9 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)
1211eleq2d 2823 . . . . . . . 8 (𝜑 → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥𝐶))
1312biimpar 481 . . . . . . 7 ((𝜑𝑥𝐶) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
14 cnextf.7 . . . . . . . 8 ((𝜑𝑥𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)
15 n0 4261 . . . . . . . 8 (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
1614, 15sylib 221 . . . . . . 7 ((𝜑𝑥𝐶) → ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
17 haustop 22228 . . . . . . . . . . . . . 14 (𝐾 ∈ Haus → 𝐾 ∈ Top)
182, 17syl 17 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ Top)
195, 6cnextfval 22959 . . . . . . . . . . . . 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 2823 . . . . . . . . . . 11 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
22 opeliunxp 5616 . . . . . . . . . . 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 290 . . . . . . . . . 10 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
2423exbidv 1929 . . . . . . . . 9 (𝜑 → (∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ ∃𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
25 19.42v 1962 . . . . . . . . 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 290 . . . . . . . 8 (𝜑 → (∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
2726biimpar 481 . . . . . . 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 502 . . . . . . 7 ((𝜑 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
3012adantr 484 . . . . . . 7 ((𝜑 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥𝐶))
3129, 30mpbid 235 . . . . . 6 ((𝜑 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)) → 𝑥𝐶)
3228, 31impbida 801 . . . . 5 (𝜑 → (𝑥𝐶 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)))
3332abbi2dv 2874 . . . 4 (𝜑𝐶 = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)})
349, 33eqtr4id 2797 . . 3 (𝜑 → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶)
35 df-fn 6383 . . 3 (((𝐽CnExt𝐾)‘𝐹) Fn 𝐶 ↔ (Fun ((𝐽CnExt𝐾)‘𝐹) ∧ dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶))
368, 34, 35sylanbrc 586 . 2 (𝜑 → ((𝐽CnExt𝐾)‘𝐹) Fn 𝐶)
3720rneqd 5807 . . 3 (𝜑 → ran ((𝐽CnExt𝐾)‘𝐹) = ran 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
38 rniun 6011 . . . 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 3412 . . . . . . . . 9 𝑥 ∈ V
4039snnz 4692 . . . . . . . 8 {𝑥} ≠ ∅
41 rnxp 6033 . . . . . . . 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 480 . . . . . . . 8 ((𝜑𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝑥𝐶)
446toptopon 21814 . . . . . . . . . . 11 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
4518, 44sylib 221 . . . . . . . . . 10 (𝜑𝐾 ∈ (TopOn‘𝐵))
4645adantr 484 . . . . . . . . 9 ((𝜑𝑥𝐶) → 𝐾 ∈ (TopOn‘𝐵))
475toptopon 21814 . . . . . . . . . . . 12 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
481, 47sylib 221 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝐶))
4948adantr 484 . . . . . . . . . 10 ((𝜑𝑥𝐶) → 𝐽 ∈ (TopOn‘𝐶))
504adantr 484 . . . . . . . . . 10 ((𝜑𝑥𝐶) → 𝐴𝐶)
51 simpr 488 . . . . . . . . . 10 ((𝜑𝑥𝐶) → 𝑥𝐶)
52 trnei 22789 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶𝑥𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
5352biimpa 480 . . . . . . . . . 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 484 . . . . . . . . 9 ((𝜑𝑥𝐶) → 𝐹:𝐴𝐵)
56 flfelbas 22891 . . . . . . . . . . 11 (((𝐾 ∈ (TopOn‘𝐵) ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴𝐵) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) → 𝑦𝐵)
5756ex 416 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝐵) ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴𝐵) → (𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) → 𝑦𝐵))
5857ssrdv 3907 . . . . . . . . 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 594 . . . . . . 7 ((𝜑𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ⊆ 𝐵)
6142, 60eqsstrid 3949 . . . . . 6 ((𝜑𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
6261ralrimiva 3105 . . . . 5 (𝜑 → ∀𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
63 iunss 4954 . . . . 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 237 . . . 4 (𝜑 𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
6538, 64eqsstrid 3949 . . 3 (𝜑 → ran 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
6637, 65eqsstrd 3939 . 2 (𝜑 → ran ((𝐽CnExt𝐾)‘𝐹) ⊆ 𝐵)
67 df-f 6384 . 2 (((𝐽CnExt𝐾)‘𝐹):𝐶𝐵 ↔ (((𝐽CnExt𝐾)‘𝐹) Fn 𝐶 ∧ ran ((𝐽CnExt𝐾)‘𝐹) ⊆ 𝐵))
6836, 66, 67sylanbrc 586 1 (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wex 1787  wcel 2110  {cab 2714  wne 2940  wral 3061  wss 3866  c0 4237  {csn 4541  cop 4547   cuni 4819   ciun 4904   × cxp 5549  dom cdm 5551  ran crn 5552  Fun wfun 6374   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  t crest 16925  Topctop 21790  TopOnctopon 21807  clsccl 21915  neicnei 21994  Hauscha 22205  Filcfil 22742   fLimf cflf 22832  CnExtccnext 22956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-map 8510  df-pm 8511  df-rest 16927  df-fbas 20360  df-fg 20361  df-top 21791  df-topon 21808  df-cld 21916  df-ntr 21917  df-cls 21918  df-nei 21995  df-haus 22212  df-fil 22743  df-fm 22835  df-flim 22836  df-flf 22837  df-cnext 22957
This theorem is referenced by:  cnextcn  22964  cnextfres1  22965
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