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Theorem flimrest 22009
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 21878 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
323adant1 1125 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
4 resttopon 21188 . . . . . 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 21874 . . . . . . . 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 21865 . . . . . . 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 21914 . . . . . . 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 22001 . . . . 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 16312 . . . . . . . . . . 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 21915 . . . . . . . . . . . 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 3744 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → ((𝑧𝑌) ∈ (𝐹t 𝑌) → (𝑧𝑌) ∈ 𝐹))
25 inss1 3977 . . . . . . . . . . . 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 20954 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → 𝑧𝑋)
2927, 28sylan 489 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧𝐽) → 𝑧𝑋)
30 filss 21879 . . . . . . . . . . . . 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 3124 . . . . . 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 3745 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝑥𝑋)
40 flimopn 22001 . . . . . . . 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 3344 . . . . . . . . 9 𝑧 ∈ V
4443inex1 4952 . . . . . . . 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 16311 . . . . . . . 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 2829 . . . . . . . . 9 (𝑦 = (𝑧𝑌) → (𝑥𝑦𝑥 ∈ (𝑧𝑌)))
50 elin 3940 . . . . . . . . . . 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 2828 . . . . . . . . 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 5032 . . . . . 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 3940 . . . 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 2759 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 2140  wral 3051  wrex 3052  Vcvv 3341  cdif 3713  cin 3715  wss 3716  cfv 6050  (class class class)co 6815  t crest 16304  fBascfbas 19957  TopOnctopon 20938  Filcfil 21871   fLim cflim 21960
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 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
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 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-oadd 7735  df-er 7914  df-en 8125  df-fin 8128  df-fi 8485  df-rest 16306  df-topgen 16327  df-fbas 19966  df-fg 19967  df-top 20922  df-topon 20939  df-bases 20973  df-ntr 21047  df-nei 21125  df-fil 21872  df-flim 21965
This theorem is referenced by:  cmetss  23334  minveclem4a  23422
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