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Theorem flimrest 23487
Description: The set of limit points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
flimrest ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ ((𝐽 β†Ύt π‘Œ) fLim (𝐹 β†Ύt π‘Œ)) = ((𝐽 fLim 𝐹) ∩ π‘Œ))
Proof of Theorem flimrest
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1137 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 filelss 23356 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ π‘Œ βŠ† 𝑋)
323adant1 1131 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ π‘Œ βŠ† 𝑋)
4 resttopon 22665 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ))
51, 3, 4syl2anc 585 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ))
6 filfbas 23352 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
763ad2ant2 1135 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
8 simp3 1139 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ π‘Œ ∈ 𝐹)
9 fbncp 23343 . . . . . . 7 ((𝐹 ∈ (fBasβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ Β¬ (𝑋 βˆ– π‘Œ) ∈ 𝐹)
107, 8, 9syl2anc 585 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ Β¬ (𝑋 βˆ– π‘Œ) ∈ 𝐹)
11 simp2 1138 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
12 trfil3 23392 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ ((𝐹 β†Ύt π‘Œ) ∈ (Filβ€˜π‘Œ) ↔ Β¬ (𝑋 βˆ– π‘Œ) ∈ 𝐹))
1311, 3, 12syl2anc 585 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ ((𝐹 β†Ύt π‘Œ) ∈ (Filβ€˜π‘Œ) ↔ Β¬ (𝑋 βˆ– π‘Œ) ∈ 𝐹))
1410, 13mpbird 257 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (𝐹 β†Ύt π‘Œ) ∈ (Filβ€˜π‘Œ))
15 flimopn 23479 . . . . 5 (((𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ) ∧ (𝐹 β†Ύt π‘Œ) ∈ (Filβ€˜π‘Œ)) β†’ (π‘₯ ∈ ((𝐽 β†Ύt π‘Œ) fLim (𝐹 β†Ύt π‘Œ)) ↔ (π‘₯ ∈ π‘Œ ∧ βˆ€π‘¦ ∈ (𝐽 β†Ύt π‘Œ)(π‘₯ ∈ 𝑦 β†’ 𝑦 ∈ (𝐹 β†Ύt π‘Œ)))))
165, 14, 15syl2anc 585 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (π‘₯ ∈ ((𝐽 β†Ύt π‘Œ) fLim (𝐹 β†Ύt π‘Œ)) ↔ (π‘₯ ∈ π‘Œ ∧ βˆ€π‘¦ ∈ (𝐽 β†Ύt π‘Œ)(π‘₯ ∈ 𝑦 β†’ 𝑦 ∈ (𝐹 β†Ύt π‘Œ)))))
17 simpll2 1214 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
18 simpll3 1215 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ π‘Œ ∈ 𝐹)
19 elrestr 17374 . . . . . . . . . . 11 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹 ∧ 𝑧 ∈ 𝐹) β†’ (𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ))
20193expia 1122 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (𝑧 ∈ 𝐹 β†’ (𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ)))
2117, 18, 20syl2anc 585 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ (𝑧 ∈ 𝐹 β†’ (𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ)))
22 trfilss 23393 . . . . . . . . . . . 12 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (𝐹 β†Ύt π‘Œ) βŠ† 𝐹)
2317, 18, 22syl2anc 585 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ (𝐹 β†Ύt π‘Œ) βŠ† 𝐹)
2423sseld 3982 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ ((𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ) β†’ (𝑧 ∩ π‘Œ) ∈ 𝐹))
25 inss1 4229 . . . . . . . . . . . 12 (𝑧 ∩ π‘Œ) βŠ† 𝑧
2625a1i 11 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ (𝑧 ∩ π‘Œ) βŠ† 𝑧)
27 simpl1 1192 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
28 toponss 22429 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝐽) β†’ 𝑧 βŠ† 𝑋)
2927, 28sylan 581 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ 𝑧 βŠ† 𝑋)
30 filss 23357 . . . . . . . . . . . . 13 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ ((𝑧 ∩ π‘Œ) ∈ 𝐹 ∧ 𝑧 βŠ† 𝑋 ∧ (𝑧 ∩ π‘Œ) βŠ† 𝑧)) β†’ 𝑧 ∈ 𝐹)
31303exp2 1355 . . . . . . . . . . . 12 (𝐹 ∈ (Filβ€˜π‘‹) β†’ ((𝑧 ∩ π‘Œ) ∈ 𝐹 β†’ (𝑧 βŠ† 𝑋 β†’ ((𝑧 ∩ π‘Œ) βŠ† 𝑧 β†’ 𝑧 ∈ 𝐹))))
3231com24 95 . . . . . . . . . . 11 (𝐹 ∈ (Filβ€˜π‘‹) β†’ ((𝑧 ∩ π‘Œ) βŠ† 𝑧 β†’ (𝑧 βŠ† 𝑋 β†’ ((𝑧 ∩ π‘Œ) ∈ 𝐹 β†’ 𝑧 ∈ 𝐹))))
3317, 26, 29, 32syl3c 66 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ ((𝑧 ∩ π‘Œ) ∈ 𝐹 β†’ 𝑧 ∈ 𝐹))
3424, 33syld 47 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ ((𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ) β†’ 𝑧 ∈ 𝐹))
3521, 34impbid 211 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ (𝑧 ∈ 𝐹 ↔ (𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ)))
3635imbi2d 341 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ ((π‘₯ ∈ 𝑧 β†’ 𝑧 ∈ 𝐹) ↔ (π‘₯ ∈ 𝑧 β†’ (𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ))))
3736ralbidva 3176 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ (βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ 𝑧 ∈ 𝐹) ↔ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ (𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ))))
38 simpl2 1193 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
393sselda 3983 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ π‘₯ ∈ 𝑋)
40 flimopn 23479 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (π‘₯ ∈ (𝐽 fLim 𝐹) ↔ (π‘₯ ∈ 𝑋 ∧ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ 𝑧 ∈ 𝐹))))
4140baibd 541 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯ ∈ (𝐽 fLim 𝐹) ↔ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ 𝑧 ∈ 𝐹)))
4227, 38, 39, 41syl21anc 837 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ (π‘₯ ∈ (𝐽 fLim 𝐹) ↔ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ 𝑧 ∈ 𝐹)))
43 vex 3479 . . . . . . . . 9 𝑧 ∈ V
4443inex1 5318 . . . . . . . 8 (𝑧 ∩ π‘Œ) ∈ V
4544a1i 11 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ (𝑧 ∩ π‘Œ) ∈ V)
46 simpl3 1194 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ π‘Œ ∈ 𝐹)
47 elrest 17373 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (𝑦 ∈ (𝐽 β†Ύt π‘Œ) ↔ βˆƒπ‘§ ∈ 𝐽 𝑦 = (𝑧 ∩ π‘Œ)))
4827, 46, 47syl2anc 585 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ (𝑦 ∈ (𝐽 β†Ύt π‘Œ) ↔ βˆƒπ‘§ ∈ 𝐽 𝑦 = (𝑧 ∩ π‘Œ)))
49 eleq2 2823 . . . . . . . . 9 (𝑦 = (𝑧 ∩ π‘Œ) β†’ (π‘₯ ∈ 𝑦 ↔ π‘₯ ∈ (𝑧 ∩ π‘Œ)))
50 elin 3965 . . . . . . . . . . 11 (π‘₯ ∈ (𝑧 ∩ π‘Œ) ↔ (π‘₯ ∈ 𝑧 ∧ π‘₯ ∈ π‘Œ))
5150rbaib 540 . . . . . . . . . 10 (π‘₯ ∈ π‘Œ β†’ (π‘₯ ∈ (𝑧 ∩ π‘Œ) ↔ π‘₯ ∈ 𝑧))
5251adantl 483 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ (π‘₯ ∈ (𝑧 ∩ π‘Œ) ↔ π‘₯ ∈ 𝑧))
5349, 52sylan9bbr 512 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑦 = (𝑧 ∩ π‘Œ)) β†’ (π‘₯ ∈ 𝑦 ↔ π‘₯ ∈ 𝑧))
54 eleq1 2822 . . . . . . . . 9 (𝑦 = (𝑧 ∩ π‘Œ) β†’ (𝑦 ∈ (𝐹 β†Ύt π‘Œ) ↔ (𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ)))
5554adantl 483 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑦 = (𝑧 ∩ π‘Œ)) β†’ (𝑦 ∈ (𝐹 β†Ύt π‘Œ) ↔ (𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ)))
5653, 55imbi12d 345 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑦 = (𝑧 ∩ π‘Œ)) β†’ ((π‘₯ ∈ 𝑦 β†’ 𝑦 ∈ (𝐹 β†Ύt π‘Œ)) ↔ (π‘₯ ∈ 𝑧 β†’ (𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ))))
5745, 48, 56ralxfr2d 5409 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ (βˆ€π‘¦ ∈ (𝐽 β†Ύt π‘Œ)(π‘₯ ∈ 𝑦 β†’ 𝑦 ∈ (𝐹 β†Ύt π‘Œ)) ↔ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ (𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ))))
5837, 42, 573bitr4d 311 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ (π‘₯ ∈ (𝐽 fLim 𝐹) ↔ βˆ€π‘¦ ∈ (𝐽 β†Ύt π‘Œ)(π‘₯ ∈ 𝑦 β†’ 𝑦 ∈ (𝐹 β†Ύt π‘Œ))))
5958pm5.32da 580 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ ((π‘₯ ∈ π‘Œ ∧ π‘₯ ∈ (𝐽 fLim 𝐹)) ↔ (π‘₯ ∈ π‘Œ ∧ βˆ€π‘¦ ∈ (𝐽 β†Ύt π‘Œ)(π‘₯ ∈ 𝑦 β†’ 𝑦 ∈ (𝐹 β†Ύt π‘Œ)))))
6016, 59bitr4d 282 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (π‘₯ ∈ ((𝐽 β†Ύt π‘Œ) fLim (𝐹 β†Ύt π‘Œ)) ↔ (π‘₯ ∈ π‘Œ ∧ π‘₯ ∈ (𝐽 fLim 𝐹))))
61 ancom 462 . . . 4 ((π‘₯ ∈ π‘Œ ∧ π‘₯ ∈ (𝐽 fLim 𝐹)) ↔ (π‘₯ ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ π‘Œ))
62 elin 3965 . . . 4 (π‘₯ ∈ ((𝐽 fLim 𝐹) ∩ π‘Œ) ↔ (π‘₯ ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ π‘Œ))
6361, 62bitr4i 278 . . 3 ((π‘₯ ∈ π‘Œ ∧ π‘₯ ∈ (𝐽 fLim 𝐹)) ↔ π‘₯ ∈ ((𝐽 fLim 𝐹) ∩ π‘Œ))
6460, 63bitrdi 287 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (π‘₯ ∈ ((𝐽 β†Ύt π‘Œ) fLim (𝐹 β†Ύt π‘Œ)) ↔ π‘₯ ∈ ((𝐽 fLim 𝐹) ∩ π‘Œ)))
6564eqrdv 2731 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ ((𝐽 β†Ύt π‘Œ) fLim (𝐹 β†Ύt π‘Œ)) = ((𝐽 fLim 𝐹) ∩ π‘Œ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  β€˜cfv 6544  (class class class)co 7409   β†Ύt crest 17366  fBascfbas 20932  TopOnctopon 22412  Filcfil 23349   fLim cflim 23438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-en 8940  df-fin 8943  df-fi 9406  df-rest 17368  df-topgen 17389  df-fbas 20941  df-fg 20942  df-top 22396  df-topon 22413  df-bases 22449  df-ntr 22524  df-nei 22602  df-fil 23350  df-flim 23443
This theorem is referenced by:  metsscmetcld  24832  cmetss  24833  minveclem4a  24947
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