MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  flimrest Structured version   Visualization version   GIF version
Theorem flimrest 23707
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 1134 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 filelss 23576 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ π‘Œ βŠ† 𝑋)
323adant1 1128 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ π‘Œ βŠ† 𝑋)
4 resttopon 22885 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ))
51, 3, 4syl2anc 582 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ))
6 filfbas 23572 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
763ad2ant2 1132 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
8 simp3 1136 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ π‘Œ ∈ 𝐹)
9 fbncp 23563 . . . . . . 7 ((𝐹 ∈ (fBasβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ Β¬ (𝑋 βˆ– π‘Œ) ∈ 𝐹)
107, 8, 9syl2anc 582 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ Β¬ (𝑋 βˆ– π‘Œ) ∈ 𝐹)
11 simp2 1135 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
12 trfil3 23612 . . . . . . 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 582 . . . . . 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 256 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (𝐹 β†Ύt π‘Œ) ∈ (Filβ€˜π‘Œ))
15 flimopn 23699 . . . . 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 582 . . . 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 1211 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
18 simpll3 1212 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ π‘Œ ∈ 𝐹)
19 elrestr 17378 . . . . . . . . . . 11 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹 ∧ 𝑧 ∈ 𝐹) β†’ (𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ))
20193expia 1119 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (𝑧 ∈ 𝐹 β†’ (𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ)))
2117, 18, 20syl2anc 582 . . . . . . . . 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 23613 . . . . . . . . . . . 12 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (𝐹 β†Ύt π‘Œ) βŠ† 𝐹)
2317, 18, 22syl2anc 582 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ (𝐹 β†Ύt π‘Œ) βŠ† 𝐹)
2423sseld 3980 . . . . . . . . . 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 4227 . . . . . . . . . . . 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 1189 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
28 toponss 22649 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝐽) β†’ 𝑧 βŠ† 𝑋)
2927, 28sylan 578 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 ∈ 𝐽) β†’ 𝑧 βŠ† 𝑋)
30 filss 23577 . . . . . . . . . . . . 13 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ ((𝑧 ∩ π‘Œ) ∈ 𝐹 ∧ 𝑧 βŠ† 𝑋 ∧ (𝑧 ∩ π‘Œ) βŠ† 𝑧)) β†’ 𝑧 ∈ 𝐹)
31303exp2 1352 . . . . . . . . . . . 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 339 . . . . . . 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 3173 . . . . . 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 1190 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
393sselda 3981 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ π‘₯ ∈ 𝑋)
40 flimopn 23699 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (π‘₯ ∈ (𝐽 fLim 𝐹) ↔ (π‘₯ ∈ 𝑋 ∧ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ 𝑧 ∈ 𝐹))))
4140baibd 538 . . . . . . 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 834 . . . . . 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 3476 . . . . . . . . 9 𝑧 ∈ V
4443inex1 5316 . . . . . . . 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 1191 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ π‘Œ ∈ 𝐹)
47 elrest 17377 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (𝑦 ∈ (𝐽 β†Ύt π‘Œ) ↔ βˆƒπ‘§ ∈ 𝐽 𝑦 = (𝑧 ∩ π‘Œ)))
4827, 46, 47syl2anc 582 . . . . . . 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 2820 . . . . . . . . 9 (𝑦 = (𝑧 ∩ π‘Œ) β†’ (π‘₯ ∈ 𝑦 ↔ π‘₯ ∈ (𝑧 ∩ π‘Œ)))
50 elin 3963 . . . . . . . . . . 11 (π‘₯ ∈ (𝑧 ∩ π‘Œ) ↔ (π‘₯ ∈ 𝑧 ∧ π‘₯ ∈ π‘Œ))
5150rbaib 537 . . . . . . . . . 10 (π‘₯ ∈ π‘Œ β†’ (π‘₯ ∈ (𝑧 ∩ π‘Œ) ↔ π‘₯ ∈ 𝑧))
5251adantl 480 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ (π‘₯ ∈ (𝑧 ∩ π‘Œ) ↔ π‘₯ ∈ 𝑧))
5349, 52sylan9bbr 509 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑦 = (𝑧 ∩ π‘Œ)) β†’ (π‘₯ ∈ 𝑦 ↔ π‘₯ ∈ 𝑧))
54 eleq1 2819 . . . . . . . . 9 (𝑦 = (𝑧 ∩ π‘Œ) β†’ (𝑦 ∈ (𝐹 β†Ύt π‘Œ) ↔ (𝑧 ∩ π‘Œ) ∈ (𝐹 β†Ύt π‘Œ)))
5554adantl 480 . . . . . . . 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 343 . . . . . . 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 5407 . . . . . 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 310 . . . . 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 577 . . . 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 281 . . 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 459 . . . 4 ((π‘₯ ∈ π‘Œ ∧ π‘₯ ∈ (𝐽 fLim 𝐹)) ↔ (π‘₯ ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ π‘Œ))
62 elin 3963 . . . 4 (π‘₯ ∈ ((𝐽 fLim 𝐹) ∩ π‘Œ) ↔ (π‘₯ ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ π‘Œ))
6361, 62bitr4i 277 . . 3 ((π‘₯ ∈ π‘Œ ∧ π‘₯ ∈ (𝐽 fLim 𝐹)) ↔ π‘₯ ∈ ((𝐽 fLim 𝐹) ∩ π‘Œ))
6460, 63bitrdi 286 . 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 2728 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   βˆ– cdif 3944   ∩ cin 3946   βŠ† wss 3947  β€˜cfv 6542  (class class class)co 7411   β†Ύt crest 17370  fBascfbas 21132  TopOnctopon 22632  Filcfil 23569   fLim cflim 23658
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-en 8942  df-fin 8945  df-fi 9408  df-rest 17372  df-topgen 17393  df-fbas 21141  df-fg 21142  df-top 22616  df-topon 22633  df-bases 22669  df-ntr 22744  df-nei 22822  df-fil 23570  df-flim 23663
This theorem is referenced by:  metsscmetcld  25063  cmetss  25064  minveclem4a  25178
  Copyright terms: Public domain W3C validator