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Theorem flimrest 21988
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 1131 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2 filelss 21857 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
323adant1 1125 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
4 resttopon 21167 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
51, 3, 4syl2anc 696 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
6 filfbas 21853 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
763ad2ant2 1129 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (fBas‘𝑋))
8 simp3 1133 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝐹)
9 fbncp 21844 . . . . . . 7 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
107, 8, 9syl2anc 696 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
11 simp2 1132 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (Fil‘𝑋))
12 trfil3 21893 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝑋) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1311, 3, 12syl2anc 696 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1410, 13mpbird 247 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐹t 𝑌) ∈ (Fil‘𝑌))
15 flimopn 21980 . . . . 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 696 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fLim (𝐹t 𝑌)) ↔ (𝑥𝑌 ∧ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦𝑦 ∈ (𝐹t 𝑌)))))
17 simpll2 1257 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → 𝐹 ∈ (Fil‘𝑋))
18 simpll3 1259 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → 𝑌𝐹)
19 elrestr 16291 . . . . . . . . . . 11 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹𝑧𝐹) → (𝑧𝑌) ∈ (𝐹t 𝑌))
20193expia 1115 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑧𝐹 → (𝑧𝑌) ∈ (𝐹t 𝑌)))
2117, 18, 20syl2anc 696 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝐹 → (𝑧𝑌) ∈ (𝐹t 𝑌)))
22 trfilss 21894 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐹t 𝑌) ⊆ 𝐹)
2317, 18, 22syl2anc 696 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝐹t 𝑌) ⊆ 𝐹)
2423sseld 3743 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → ((𝑧𝑌) ∈ (𝐹t 𝑌) → (𝑧𝑌) ∈ 𝐹))
25 inss1 3976 . . . . . . . . . . . 12 (𝑧𝑌) ⊆ 𝑧
2625a1i 11 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝑌) ⊆ 𝑧)
27 simpl1 1228 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐽 ∈ (TopOn‘𝑋))
28 toponss 20933 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → 𝑧𝑋)
2927, 28sylan 489 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → 𝑧𝑋)
30 filss 21858 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝑧𝑌) ∈ 𝐹𝑧𝑋 ∧ (𝑧𝑌) ⊆ 𝑧)) → 𝑧𝐹)
31303exp2 1448 . . . . . . . . . . . 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 202 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝐹 ↔ (𝑧𝑌) ∈ (𝐹t 𝑌)))
3635imbi2d 329 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → ((𝑥𝑧𝑧𝐹) ↔ (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
3736ralbidva 3123 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑧𝐽 (𝑥𝑧𝑧𝐹) ↔ ∀𝑧𝐽 (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
38 simpl2 1230 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐹 ∈ (Fil‘𝑋))
393sselda 3744 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝑥𝑋)
40 flimopn 21980 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑧𝐽 (𝑥𝑧𝑧𝐹))))
4140baibd 986 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝑋) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑧𝐽 (𝑥𝑧𝑧𝐹)))
4227, 38, 39, 41syl21anc 1476 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑧𝐽 (𝑥𝑧𝑧𝐹)))
43 vex 3343 . . . . . . . . 9 𝑧 ∈ V
4443inex1 4951 . . . . . . . 8 (𝑧𝑌) ∈ V
4544a1i 11 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → (𝑧𝑌) ∈ V)
46 simpl3 1232 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝑌𝐹)
47 elrest 16290 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝐹) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑧𝐽 𝑦 = (𝑧𝑌)))
4827, 46, 47syl2anc 696 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑧𝐽 𝑦 = (𝑧𝑌)))
49 eleq2 2828 . . . . . . . . 9 (𝑦 = (𝑧𝑌) → (𝑥𝑦𝑥 ∈ (𝑧𝑌)))
50 elin 3939 . . . . . . . . . . 11 (𝑥 ∈ (𝑧𝑌) ↔ (𝑥𝑧𝑥𝑌))
5150rbaib 985 . . . . . . . . . 10 (𝑥𝑌 → (𝑥 ∈ (𝑧𝑌) ↔ 𝑥𝑧))
5251adantl 473 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝑧𝑌) ↔ 𝑥𝑧))
5349, 52sylan9bbr 739 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑧𝑌)) → (𝑥𝑦𝑥𝑧))
54 eleq1 2827 . . . . . . . . 9 (𝑦 = (𝑧𝑌) → (𝑦 ∈ (𝐹t 𝑌) ↔ (𝑧𝑌) ∈ (𝐹t 𝑌)))
5554adantl 473 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑧𝑌)) → (𝑦 ∈ (𝐹t 𝑌) ↔ (𝑧𝑌) ∈ (𝐹t 𝑌)))
5653, 55imbi12d 333 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑧𝑌)) → ((𝑥𝑦𝑦 ∈ (𝐹t 𝑌)) ↔ (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
5745, 48, 56ralxfr2d 5031 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦𝑦 ∈ (𝐹t 𝑌)) ↔ ∀𝑧𝐽 (𝑥𝑧 → (𝑧𝑌) ∈ (𝐹t 𝑌))))
5837, 42, 573bitr4d 300 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦𝑦 ∈ (𝐹t 𝑌))))
5958pm5.32da 676 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹)) ↔ (𝑥𝑌 ∧ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦𝑦 ∈ (𝐹t 𝑌)))))
6016, 59bitr4d 271 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fLim (𝐹t 𝑌)) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹))))
61 ancom 465 . . . 4 ((𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹)) ↔ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑥𝑌))
62 elin 3939 . . . 4 (𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑥𝑌))
6361, 62bitr4i 267 . . 3 ((𝑥𝑌𝑥 ∈ (𝐽 fLim 𝐹)) ↔ 𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌))
6460, 63syl6bb 276 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fLim (𝐹t 𝑌)) ↔ 𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌)))
6564eqrdv 2758 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  cdif 3712  cin 3714  wss 3715  cfv 6049  (class class class)co 6813  t crest 16283  fBascfbas 19936  TopOnctopon 20917  Filcfil 21850   fLim cflim 21939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-oadd 7733  df-er 7911  df-en 8122  df-fin 8125  df-fi 8482  df-rest 16285  df-topgen 16306  df-fbas 19945  df-fg 19946  df-top 20901  df-topon 20918  df-bases 20952  df-ntr 21026  df-nei 21104  df-fil 21851  df-flim 21944
This theorem is referenced by:  cmetss  23313  minveclem4a  23401
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