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Theorem flimrest 22510
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 1130 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2 filelss 22379 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
323adant1 1124 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
4 resttopon 21688 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
51, 3, 4syl2anc 584 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
6 filfbas 22375 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
763ad2ant2 1128 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (fBas‘𝑋))
8 simp3 1132 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝐹)
9 fbncp 22366 . . . . . . 7 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
107, 8, 9syl2anc 584 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
11 simp2 1131 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (Fil‘𝑋))
12 trfil3 22415 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝑋) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1311, 3, 12syl2anc 584 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1410, 13mpbird 258 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐹t 𝑌) ∈ (Fil‘𝑌))
15 flimopn 22502 . . . . 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 584 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fLim (𝐹t 𝑌)) ↔ (𝑥𝑌 ∧ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦𝑦 ∈ (𝐹t 𝑌)))))
17 simpll2 1207 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → 𝐹 ∈ (Fil‘𝑋))
18 simpll3 1208 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → 𝑌𝐹)
19 elrestr 16692 . . . . . . . . . . 11 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹𝑧𝐹) → (𝑧𝑌) ∈ (𝐹t 𝑌))
20193expia 1115 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑧𝐹 → (𝑧𝑌) ∈ (𝐹t 𝑌)))
2117, 18, 20syl2anc 584 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝐹 → (𝑧𝑌) ∈ (𝐹t 𝑌)))
22 trfilss 22416 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐹t 𝑌) ⊆ 𝐹)
2317, 18, 22syl2anc 584 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝐹t 𝑌) ⊆ 𝐹)
2423sseld 3970 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → ((𝑧𝑌) ∈ (𝐹t 𝑌) → (𝑧𝑌) ∈ 𝐹))
25 inss1 4209 . . . . . . . . . . . 12 (𝑧𝑌) ⊆ 𝑧
2625a1i 11 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝑌) ⊆ 𝑧)
27 simpl1 1185 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐽 ∈ (TopOn‘𝑋))
28 toponss 21454 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → 𝑧𝑋)
2927, 28sylan 580 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → 𝑧𝑋)
30 filss 22380 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝑧𝑌) ∈ 𝐹𝑧𝑋 ∧ (𝑧𝑌) ⊆ 𝑧)) → 𝑧𝐹)
31303exp2 1348 . . . . . . . . . . . 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 213 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝐹 ↔ (𝑧𝑌) ∈ (𝐹t 𝑌)))
3635imbi2d 342 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → ((𝑥𝑧𝑧𝐹) ↔ (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
3736ralbidva 3201 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑧𝐽 (𝑥𝑧𝑧𝐹) ↔ ∀𝑧𝐽 (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
38 simpl2 1186 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐹 ∈ (Fil‘𝑋))
393sselda 3971 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝑥𝑋)
40 flimopn 22502 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑧𝐽 (𝑥𝑧𝑧𝐹))))
4140baibd 540 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝑋) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑧𝐽 (𝑥𝑧𝑧𝐹)))
4227, 38, 39, 41syl21anc 835 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑧𝐽 (𝑥𝑧𝑧𝐹)))
43 vex 3503 . . . . . . . . 9 𝑧 ∈ V
4443inex1 5218 . . . . . . . 8 (𝑧𝑌) ∈ V
4544a1i 11 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝑌) ∈ V)
46 simpl3 1187 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝑌𝐹)
47 elrest 16691 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝐹) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑧𝐽 𝑦 = (𝑧𝑌)))
4827, 46, 47syl2anc 584 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑧𝐽 𝑦 = (𝑧𝑌)))
49 eleq2 2906 . . . . . . . . 9 (𝑦 = (𝑧𝑌) → (𝑥𝑦𝑥 ∈ (𝑧𝑌)))
50 elin 4173 . . . . . . . . . . 11 (𝑥 ∈ (𝑧𝑌) ↔ (𝑥𝑧𝑥𝑌))
5150rbaib 539 . . . . . . . . . 10 (𝑥𝑌 → (𝑥 ∈ (𝑧𝑌) ↔ 𝑥𝑧))
5251adantl 482 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝑧𝑌) ↔ 𝑥𝑧))
5349, 52sylan9bbr 511 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑧𝑌)) → (𝑥𝑦𝑥𝑧))
54 eleq1 2905 . . . . . . . . 9 (𝑦 = (𝑧𝑌) → (𝑦 ∈ (𝐹t 𝑌) ↔ (𝑧𝑌) ∈ (𝐹t 𝑌)))
5554adantl 482 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑧𝑌)) → (𝑦 ∈ (𝐹t 𝑌) ↔ (𝑧𝑌) ∈ (𝐹t 𝑌)))
5653, 55imbi12d 346 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑧𝑌)) → ((𝑥𝑦𝑦 ∈ (𝐹t 𝑌)) ↔ (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
5745, 48, 56ralxfr2d 5307 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦𝑦 ∈ (𝐹t 𝑌)) ↔ ∀𝑧𝐽 (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
5837, 42, 573bitr4d 312 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦𝑦 ∈ (𝐹t 𝑌))))
5958pm5.32da 579 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹)) ↔ (𝑥𝑌 ∧ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦𝑦 ∈ (𝐹t 𝑌)))))
6016, 59bitr4d 283 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fLim (𝐹t 𝑌)) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹))))
61 ancom 461 . . . 4 ((𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹)) ↔ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑥𝑌))
62 elin 4173 . . . 4 (𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑥𝑌))
6361, 62bitr4i 279 . . 3 ((𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹)) ↔ 𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌))
6460, 63syl6bb 288 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fLim (𝐹t 𝑌)) ↔ 𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌)))
6564eqrdv 2824 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3143  wrex 3144  Vcvv 3500  cdif 3937  cin 3939  wss 3940  cfv 6352  (class class class)co 7148  t crest 16684  fBascfbas 20452  TopOnctopon 21437  Filcfil 22372   fLim cflim 22461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-oadd 8097  df-er 8279  df-en 8499  df-fin 8502  df-fi 8864  df-rest 16686  df-topgen 16707  df-fbas 20461  df-fg 20462  df-top 21421  df-topon 21438  df-bases 21473  df-ntr 21547  df-nei 21625  df-fil 22373  df-flim 22466
This theorem is referenced by:  metsscmetcld  23836  cmetss  23837  minveclem4a  23951
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