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Theorem flimrest 22157
Description: The set of limit points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
flimrest ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝐽t 𝑌) fLim (𝐹t 𝑌)) = ((𝐽 fLim 𝐹) ∩ 𝑌))

Proof of Theorem flimrest
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1170 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2 filelss 22026 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
323adant1 1164 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
4 resttopon 21336 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
51, 3, 4syl2anc 579 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
6 filfbas 22022 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
763ad2ant2 1168 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (fBas‘𝑋))
8 simp3 1172 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝐹)
9 fbncp 22013 . . . . . . 7 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
107, 8, 9syl2anc 579 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
11 simp2 1171 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (Fil‘𝑋))
12 trfil3 22062 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝑋) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1311, 3, 12syl2anc 579 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1410, 13mpbird 249 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐹t 𝑌) ∈ (Fil‘𝑌))
15 flimopn 22149 . . . . 5 (((𝐽t 𝑌) ∈ (TopOn‘𝑌) ∧ (𝐹t 𝑌) ∈ (Fil‘𝑌)) → (𝑥 ∈ ((𝐽t 𝑌) fLim (𝐹t 𝑌)) ↔ (𝑥𝑌 ∧ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦𝑦 ∈ (𝐹t 𝑌)))))
165, 14, 15syl2anc 579 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fLim (𝐹t 𝑌)) ↔ (𝑥𝑌 ∧ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦𝑦 ∈ (𝐹t 𝑌)))))
17 simpll2 1275 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → 𝐹 ∈ (Fil‘𝑋))
18 simpll3 1277 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → 𝑌𝐹)
19 elrestr 16442 . . . . . . . . . . 11 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹𝑧𝐹) → (𝑧𝑌) ∈ (𝐹t 𝑌))
20193expia 1154 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑧𝐹 → (𝑧𝑌) ∈ (𝐹t 𝑌)))
2117, 18, 20syl2anc 579 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝐹 → (𝑧𝑌) ∈ (𝐹t 𝑌)))
22 trfilss 22063 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐹t 𝑌) ⊆ 𝐹)
2317, 18, 22syl2anc 579 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝐹t 𝑌) ⊆ 𝐹)
2423sseld 3826 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → ((𝑧𝑌) ∈ (𝐹t 𝑌) → (𝑧𝑌) ∈ 𝐹))
25 inss1 4057 . . . . . . . . . . . 12 (𝑧𝑌) ⊆ 𝑧
2625a1i 11 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝑌) ⊆ 𝑧)
27 simpl1 1246 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐽 ∈ (TopOn‘𝑋))
28 toponss 21102 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → 𝑧𝑋)
2927, 28sylan 575 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → 𝑧𝑋)
30 filss 22027 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝑧𝑌) ∈ 𝐹𝑧𝑋 ∧ (𝑧𝑌) ⊆ 𝑧)) → 𝑧𝐹)
31303exp2 1467 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → ((𝑧𝑌) ∈ 𝐹 → (𝑧𝑋 → ((𝑧𝑌) ⊆ 𝑧𝑧𝐹))))
3231com24 95 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → ((𝑧𝑌) ⊆ 𝑧 → (𝑧𝑋 → ((𝑧𝑌) ∈ 𝐹𝑧𝐹))))
3317, 26, 29, 32syl3c 66 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → ((𝑧𝑌) ∈ 𝐹𝑧𝐹))
3424, 33syld 47 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → ((𝑧𝑌) ∈ (𝐹t 𝑌) → 𝑧𝐹))
3521, 34impbid 204 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝐹 ↔ (𝑧𝑌) ∈ (𝐹t 𝑌)))
3635imbi2d 332 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → ((𝑥𝑧𝑧𝐹) ↔ (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
3736ralbidva 3194 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑧𝐽 (𝑥𝑧𝑧𝐹) ↔ ∀𝑧𝐽 (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
38 simpl2 1248 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐹 ∈ (Fil‘𝑋))
393sselda 3827 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝑥𝑋)
40 flimopn 22149 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑧𝐽 (𝑥𝑧𝑧𝐹))))
4140baibd 535 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝑋) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑧𝐽 (𝑥𝑧𝑧𝐹)))
4227, 38, 39, 41syl21anc 871 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑧𝐽 (𝑥𝑧𝑧𝐹)))
43 vex 3417 . . . . . . . . 9 𝑧 ∈ V
4443inex1 5024 . . . . . . . 8 (𝑧𝑌) ∈ V
4544a1i 11 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝑌) ∈ V)
46 simpl3 1250 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝑌𝐹)
47 elrest 16441 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝐹) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑧𝐽 𝑦 = (𝑧𝑌)))
4827, 46, 47syl2anc 579 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑧𝐽 𝑦 = (𝑧𝑌)))
49 eleq2 2895 . . . . . . . . 9 (𝑦 = (𝑧𝑌) → (𝑥𝑦𝑥 ∈ (𝑧𝑌)))
50 elin 4023 . . . . . . . . . . 11 (𝑥 ∈ (𝑧𝑌) ↔ (𝑥𝑧𝑥𝑌))
5150rbaib 534 . . . . . . . . . 10 (𝑥𝑌 → (𝑥 ∈ (𝑧𝑌) ↔ 𝑥𝑧))
5251adantl 475 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝑧𝑌) ↔ 𝑥𝑧))
5349, 52sylan9bbr 506 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑧𝑌)) → (𝑥𝑦𝑥𝑧))
54 eleq1 2894 . . . . . . . . 9 (𝑦 = (𝑧𝑌) → (𝑦 ∈ (𝐹t 𝑌) ↔ (𝑧𝑌) ∈ (𝐹t 𝑌)))
5554adantl 475 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑧𝑌)) → (𝑦 ∈ (𝐹t 𝑌) ↔ (𝑧𝑌) ∈ (𝐹t 𝑌)))
5653, 55imbi12d 336 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑧𝑌)) → ((𝑥𝑦𝑦 ∈ (𝐹t 𝑌)) ↔ (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
5745, 48, 56ralxfr2d 5110 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦𝑦 ∈ (𝐹t 𝑌)) ↔ ∀𝑧𝐽 (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
5837, 42, 573bitr4d 303 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦𝑦 ∈ (𝐹t 𝑌))))
5958pm5.32da 574 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹)) ↔ (𝑥𝑌 ∧ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦𝑦 ∈ (𝐹t 𝑌)))))
6016, 59bitr4d 274 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fLim (𝐹t 𝑌)) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹))))
61 ancom 454 . . . 4 ((𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹)) ↔ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑥𝑌))
62 elin 4023 . . . 4 (𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑥𝑌))
6361, 62bitr4i 270 . . 3 ((𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹)) ↔ 𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌))
6460, 63syl6bb 279 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fLim (𝐹t 𝑌)) ↔ 𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌)))
6564eqrdv 2823 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝐽t 𝑌) fLim (𝐹t 𝑌)) = ((𝐽 fLim 𝐹) ∩ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wral 3117  wrex 3118  Vcvv 3414  cdif 3795  cin 3797  wss 3798  cfv 6123  (class class class)co 6905  t crest 16434  fBascfbas 20094  TopOnctopon 21085  Filcfil 22019   fLim cflim 22108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-oadd 7830  df-er 8009  df-en 8223  df-fin 8226  df-fi 8586  df-rest 16436  df-topgen 16457  df-fbas 20103  df-fg 20104  df-top 21069  df-topon 21086  df-bases 21121  df-ntr 21195  df-nei 21273  df-fil 22020  df-flim 22113
This theorem is referenced by:  metsscmetcld  23483  cmetss  23484  minveclem4a  23598
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