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Theorem flimrest 23991
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 1137 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2 filelss 23860 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
323adant1 1131 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
4 resttopon 23169 . . . . . 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 23856 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
763ad2ant2 1135 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (fBas‘𝑋))
8 simp3 1139 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝐹)
9 fbncp 23847 . . . . . . 7 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
107, 8, 9syl2anc 584 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
11 simp2 1138 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (Fil‘𝑋))
12 trfil3 23896 . . . . . . 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 257 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐹t 𝑌) ∈ (Fil‘𝑌))
15 flimopn 23983 . . . . 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 1214 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → 𝐹 ∈ (Fil‘𝑋))
18 simpll3 1215 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → 𝑌𝐹)
19 elrestr 17473 . . . . . . . . . . 11 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹𝑧𝐹) → (𝑧𝑌) ∈ (𝐹t 𝑌))
20193expia 1122 . . . . . . . . . 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 23897 . . . . . . . . . . . 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 3982 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → ((𝑧𝑌) ∈ (𝐹t 𝑌) → (𝑧𝑌) ∈ 𝐹))
25 inss1 4237 . . . . . . . . . . . 12 (𝑧𝑌) ⊆ 𝑧
2625a1i 11 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝑌) ⊆ 𝑧)
27 simpl1 1192 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐽 ∈ (TopOn‘𝑋))
28 toponss 22933 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → 𝑧𝑋)
2927, 28sylan 580 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → 𝑧𝑋)
30 filss 23861 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝑧𝑌) ∈ 𝐹𝑧𝑋 ∧ (𝑧𝑌) ⊆ 𝑧)) → 𝑧𝐹)
31303exp2 1355 . . . . . . . . . . . 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 212 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝐹 ↔ (𝑧𝑌) ∈ (𝐹t 𝑌)))
3635imbi2d 340 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → ((𝑥𝑧𝑧𝐹) ↔ (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
3736ralbidva 3176 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑧𝐽 (𝑥𝑧𝑧𝐹) ↔ ∀𝑧𝐽 (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
38 simpl2 1193 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐹 ∈ (Fil‘𝑋))
393sselda 3983 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝑥𝑋)
40 flimopn 23983 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑧𝐽 (𝑥𝑧𝑧𝐹))))
4140baibd 539 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝑋) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑧𝐽 (𝑥𝑧𝑧𝐹)))
4227, 38, 39, 41syl21anc 838 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑧𝐽 (𝑥𝑧𝑧𝐹)))
43 vex 3484 . . . . . . . . 9 𝑧 ∈ V
4443inex1 5317 . . . . . . . 8 (𝑧𝑌) ∈ V
4544a1i 11 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝑌) ∈ V)
46 simpl3 1194 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝑌𝐹)
47 elrest 17472 . . . . . . . 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 2830 . . . . . . . . 9 (𝑦 = (𝑧𝑌) → (𝑥𝑦𝑥 ∈ (𝑧𝑌)))
50 elin 3967 . . . . . . . . . . 11 (𝑥 ∈ (𝑧𝑌) ↔ (𝑥𝑧𝑥𝑌))
5150rbaib 538 . . . . . . . . . 10 (𝑥𝑌 → (𝑥 ∈ (𝑧𝑌) ↔ 𝑥𝑧))
5251adantl 481 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝑧𝑌) ↔ 𝑥𝑧))
5349, 52sylan9bbr 510 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑧𝑌)) → (𝑥𝑦𝑥𝑧))
54 eleq1 2829 . . . . . . . . 9 (𝑦 = (𝑧𝑌) → (𝑦 ∈ (𝐹t 𝑌) ↔ (𝑧𝑌) ∈ (𝐹t 𝑌)))
5554adantl 481 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑧𝑌)) → (𝑦 ∈ (𝐹t 𝑌) ↔ (𝑧𝑌) ∈ (𝐹t 𝑌)))
5653, 55imbi12d 344 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑧𝑌)) → ((𝑥𝑦𝑦 ∈ (𝐹t 𝑌)) ↔ (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
5745, 48, 56ralxfr2d 5410 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦𝑦 ∈ (𝐹t 𝑌)) ↔ ∀𝑧𝐽 (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
5837, 42, 573bitr4d 311 . . . . 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 282 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fLim (𝐹t 𝑌)) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹))))
61 ancom 460 . . . 4 ((𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹)) ↔ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑥𝑌))
62 elin 3967 . . . 4 (𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑥𝑌))
6361, 62bitr4i 278 . . 3 ((𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹)) ↔ 𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌))
6460, 63bitrdi 287 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fLim (𝐹t 𝑌)) ↔ 𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌)))
6564eqrdv 2735 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  cin 3950  wss 3951  cfv 6561  (class class class)co 7431  t crest 17465  fBascfbas 21352  TopOnctopon 22916  Filcfil 23853   fLim cflim 23942
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-3or 1088  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-pss 3971  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-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  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-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  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-om 7888  df-1st 8014  df-2nd 8015  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-fbas 21361  df-fg 21362  df-top 22900  df-topon 22917  df-bases 22953  df-ntr 23028  df-nei 23106  df-fil 23854  df-flim 23947
This theorem is referenced by:  metsscmetcld  25349  cmetss  25350  minveclem4a  25464
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