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Theorem cnextfun 22067
Description: If the target space is Hausdorff, a continuous extension is a function. (Contributed by Thierry Arnoux, 20-Dec-2017.)
Hypotheses
Ref Expression
cnextfrel.1 𝐶 = 𝐽
cnextfrel.2 𝐵 = 𝐾
Assertion
Ref Expression
cnextfun (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹))

Proof of Theorem cnextfun
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 haustop 21335 . . 3 (𝐾 ∈ Haus → 𝐾 ∈ Top)
2 cnextfrel.1 . . . 4 𝐶 = 𝐽
3 cnextfrel.2 . . . 4 𝐵 = 𝐾
42, 3cnextrel 22066 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → Rel ((𝐽CnExt𝐾)‘𝐹))
51, 4sylanl2 686 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → Rel ((𝐽CnExt𝐾)‘𝐹))
6 simpllr 817 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝐾 ∈ Haus)
72toptopon 20922 . . . . . . . . . 10 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
87biimpi 206 . . . . . . . . 9 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝐶))
98ad3antrrr 768 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝐽 ∈ (TopOn‘𝐶))
10 simplrr 820 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝐴𝐶)
119, 7sylibr 224 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝐽 ∈ Top)
122clsss3 21063 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐴𝐶) → ((cls‘𝐽)‘𝐴) ⊆ 𝐶)
1311, 10, 12syl2anc 696 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ((cls‘𝐽)‘𝐴) ⊆ 𝐶)
14 simpr 479 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
1513, 14sseldd 3743 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝑥𝐶)
16 trnei 21895 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶𝑥𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
1716biimpa 502 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶𝑥𝐶) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
189, 10, 15, 14, 17syl31anc 1480 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
19 simplrl 819 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝐹:𝐴𝐵)
203hausflf 22000 . . . . . . 7 ((𝐾 ∈ Haus ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴𝐵) → ∃*𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
216, 18, 19, 20syl3anc 1477 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ∃*𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
2221ex 449 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) → ∃*𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
2322alrimiv 2002 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → ∀𝑥(𝑥 ∈ ((cls‘𝐽)‘𝐴) → ∃*𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
24 moanimv 2667 . . . . 5 (∃*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)))
2524albii 1894 . . . 4 (∀𝑥∃*𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ ∀𝑥(𝑥 ∈ ((cls‘𝐽)‘𝐴) → ∃*𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
2623, 25sylibr 224 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → ∀𝑥∃*𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
27 df-br 4803 . . . . . . 7 (𝑥((𝐽CnExt𝐾)‘𝐹)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹))
2827a1i 11 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → (𝑥((𝐽CnExt𝐾)‘𝐹)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)))
292, 3cnextfval 22065 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → ((𝐽CnExt𝐾)‘𝐹) = 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
301, 29sylanl2 686 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → ((𝐽CnExt𝐾)‘𝐹) = 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
3130eleq2d 2823 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → (⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
32 opeliunxp 5325 . . . . . . 7 (⟨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)))
3332a1i 11 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → (⟨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))))
3428, 31, 333bitrd 294 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → (𝑥((𝐽CnExt𝐾)‘𝐹)𝑦 ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
3534mobidv 2626 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → (∃*𝑦 𝑥((𝐽CnExt𝐾)‘𝐹)𝑦 ↔ ∃*𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
3635albidv 1996 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → (∀𝑥∃*𝑦 𝑥((𝐽CnExt𝐾)‘𝐹)𝑦 ↔ ∀𝑥∃*𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
3726, 36mpbird 247 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → ∀𝑥∃*𝑦 𝑥((𝐽CnExt𝐾)‘𝐹)𝑦)
38 dffun6 6062 . 2 (Fun ((𝐽CnExt𝐾)‘𝐹) ↔ (Rel ((𝐽CnExt𝐾)‘𝐹) ∧ ∀𝑥∃*𝑦 𝑥((𝐽CnExt𝐾)‘𝐹)𝑦))
395, 37, 38sylanbrc 701 1 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wal 1628   = wceq 1630  wcel 2137  ∃*wmo 2606  wss 3713  {csn 4319  cop 4325   cuni 4586   ciun 4670   class class class wbr 4802   × cxp 5262  Rel wrel 5269  Fun wfun 6041  wf 6043  cfv 6047  (class class class)co 6811  t crest 16281  Topctop 20898  TopOnctopon 20915  clsccl 21022  neicnei 21101  Hauscha 21312  Filcfil 21848   fLimf cflf 21938  CnExtccnext 22062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-iin 4673  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-1st 7331  df-2nd 7332  df-map 8023  df-pm 8024  df-rest 16283  df-fbas 19943  df-top 20899  df-topon 20916  df-cld 21023  df-ntr 21024  df-cls 21025  df-nei 21102  df-haus 21319  df-fil 21849  df-flim 21942  df-flf 21943  df-cnext 22063
This theorem is referenced by:  cnextfvval  22068  cnextf  22069  cnextfres  22072
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