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Theorem fclsrest 23528
Description: The set of cluster points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
fclsrest ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ ((𝐽 β†Ύt π‘Œ) fClus (𝐹 β†Ύt π‘Œ)) = ((𝐽 fClus 𝐹) ∩ π‘Œ))
Proof of Theorem fclsrest
Dummy variables 𝑠 𝑑 𝑒 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 fclsopn 23518 . . . . 5 (((𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ) ∧ (𝐹 β†Ύt π‘Œ) ∈ (Filβ€˜π‘Œ)) β†’ (π‘₯ ∈ ((𝐽 β†Ύt π‘Œ) fClus (𝐹 β†Ύt π‘Œ)) ↔ (π‘₯ ∈ π‘Œ ∧ βˆ€π‘¦ ∈ (𝐽 β†Ύt π‘Œ)(π‘₯ ∈ 𝑦 β†’ βˆ€π‘§ ∈ (𝐹 β†Ύt π‘Œ)(𝑦 ∩ 𝑧) β‰  βˆ…))))
165, 14, 15syl2anc 585 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (π‘₯ ∈ ((𝐽 β†Ύt π‘Œ) fClus (𝐹 β†Ύt π‘Œ)) ↔ (π‘₯ ∈ π‘Œ ∧ βˆ€π‘¦ ∈ (𝐽 β†Ύt π‘Œ)(π‘₯ ∈ 𝑦 β†’ βˆ€π‘§ ∈ (𝐹 β†Ύt π‘Œ)(𝑦 ∩ 𝑧) β‰  βˆ…))))
17 in32 4222 . . . . . . . . . . . . . 14 ((𝑒 ∩ 𝑠) ∩ π‘Œ) = ((𝑒 ∩ π‘Œ) ∩ 𝑠)
18 ineq2 4207 . . . . . . . . . . . . . 14 (𝑠 = 𝑑 β†’ ((𝑒 ∩ π‘Œ) ∩ 𝑠) = ((𝑒 ∩ π‘Œ) ∩ 𝑑))
1917, 18eqtrid 2785 . . . . . . . . . . . . 13 (𝑠 = 𝑑 β†’ ((𝑒 ∩ 𝑠) ∩ π‘Œ) = ((𝑒 ∩ π‘Œ) ∩ 𝑑))
2019neeq1d 3001 . . . . . . . . . . . 12 (𝑠 = 𝑑 β†’ (((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ… ↔ ((𝑒 ∩ π‘Œ) ∩ 𝑑) β‰  βˆ…))
2120rspccv 3610 . . . . . . . . . . 11 (βˆ€π‘  ∈ 𝐹 ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ… β†’ (𝑑 ∈ 𝐹 β†’ ((𝑒 ∩ π‘Œ) ∩ 𝑑) β‰  βˆ…))
22 inss1 4229 . . . . . . . . . . . . 13 (𝑒 ∩ π‘Œ) βŠ† 𝑒
23 ssrin 4234 . . . . . . . . . . . . 13 ((𝑒 ∩ π‘Œ) βŠ† 𝑒 β†’ ((𝑒 ∩ π‘Œ) ∩ 𝑑) βŠ† (𝑒 ∩ 𝑑))
2422, 23ax-mp 5 . . . . . . . . . . . 12 ((𝑒 ∩ π‘Œ) ∩ 𝑑) βŠ† (𝑒 ∩ 𝑑)
25 ssn0 4401 . . . . . . . . . . . 12 ((((𝑒 ∩ π‘Œ) ∩ 𝑑) βŠ† (𝑒 ∩ 𝑑) ∧ ((𝑒 ∩ π‘Œ) ∩ 𝑑) β‰  βˆ…) β†’ (𝑒 ∩ 𝑑) β‰  βˆ…)
2624, 25mpan 689 . . . . . . . . . . 11 (((𝑒 ∩ π‘Œ) ∩ 𝑑) β‰  βˆ… β†’ (𝑒 ∩ 𝑑) β‰  βˆ…)
2721, 26syl6 35 . . . . . . . . . 10 (βˆ€π‘  ∈ 𝐹 ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ… β†’ (𝑑 ∈ 𝐹 β†’ (𝑒 ∩ 𝑑) β‰  βˆ…))
2827ralrimiv 3146 . . . . . . . . 9 (βˆ€π‘  ∈ 𝐹 ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ… β†’ βˆ€π‘‘ ∈ 𝐹 (𝑒 ∩ 𝑑) β‰  βˆ…)
2911ad3antrrr 729 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑒 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
30 simpr 486 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑒 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) β†’ 𝑠 ∈ 𝐹)
318ad3antrrr 729 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑒 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) β†’ π‘Œ ∈ 𝐹)
32 filin 23358 . . . . . . . . . . . 12 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑠 ∈ 𝐹 ∧ π‘Œ ∈ 𝐹) β†’ (𝑠 ∩ π‘Œ) ∈ 𝐹)
3329, 30, 31, 32syl3anc 1372 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑒 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) β†’ (𝑠 ∩ π‘Œ) ∈ 𝐹)
34 ineq2 4207 . . . . . . . . . . . . . 14 (𝑑 = (𝑠 ∩ π‘Œ) β†’ (𝑒 ∩ 𝑑) = (𝑒 ∩ (𝑠 ∩ π‘Œ)))
35 inass 4220 . . . . . . . . . . . . . 14 ((𝑒 ∩ 𝑠) ∩ π‘Œ) = (𝑒 ∩ (𝑠 ∩ π‘Œ))
3634, 35eqtr4di 2791 . . . . . . . . . . . . 13 (𝑑 = (𝑠 ∩ π‘Œ) β†’ (𝑒 ∩ 𝑑) = ((𝑒 ∩ 𝑠) ∩ π‘Œ))
3736neeq1d 3001 . . . . . . . . . . . 12 (𝑑 = (𝑠 ∩ π‘Œ) β†’ ((𝑒 ∩ 𝑑) β‰  βˆ… ↔ ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ…))
3837rspcv 3609 . . . . . . . . . . 11 ((𝑠 ∩ π‘Œ) ∈ 𝐹 β†’ (βˆ€π‘‘ ∈ 𝐹 (𝑒 ∩ 𝑑) β‰  βˆ… β†’ ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ…))
3933, 38syl 17 . . . . . . . . . 10 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑒 ∈ 𝐽) ∧ 𝑠 ∈ 𝐹) β†’ (βˆ€π‘‘ ∈ 𝐹 (𝑒 ∩ 𝑑) β‰  βˆ… β†’ ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ…))
4039ralrimdva 3155 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑒 ∈ 𝐽) β†’ (βˆ€π‘‘ ∈ 𝐹 (𝑒 ∩ 𝑑) β‰  βˆ… β†’ βˆ€π‘  ∈ 𝐹 ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ…))
4128, 40impbid2 225 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑒 ∈ 𝐽) β†’ (βˆ€π‘  ∈ 𝐹 ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ… ↔ βˆ€π‘‘ ∈ 𝐹 (𝑒 ∩ 𝑑) β‰  βˆ…))
4241imbi2d 341 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑒 ∈ 𝐽) β†’ ((π‘₯ ∈ 𝑒 β†’ βˆ€π‘  ∈ 𝐹 ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ…) ↔ (π‘₯ ∈ 𝑒 β†’ βˆ€π‘‘ ∈ 𝐹 (𝑒 ∩ 𝑑) β‰  βˆ…)))
4342ralbidva 3176 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ (βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆ€π‘  ∈ 𝐹 ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ…) ↔ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆ€π‘‘ ∈ 𝐹 (𝑒 ∩ 𝑑) β‰  βˆ…)))
44 vex 3479 . . . . . . . . 9 𝑒 ∈ V
4544inex1 5318 . . . . . . . 8 (𝑒 ∩ π‘Œ) ∈ V
4645a1i 11 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑒 ∈ 𝐽) β†’ (𝑒 ∩ π‘Œ) ∈ V)
47 elrest 17373 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (𝑦 ∈ (𝐽 β†Ύt π‘Œ) ↔ βˆƒπ‘’ ∈ 𝐽 𝑦 = (𝑒 ∩ π‘Œ)))
48473adant2 1132 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (𝑦 ∈ (𝐽 β†Ύt π‘Œ) ↔ βˆƒπ‘’ ∈ 𝐽 𝑦 = (𝑒 ∩ π‘Œ)))
4948adantr 482 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ (𝑦 ∈ (𝐽 β†Ύt π‘Œ) ↔ βˆƒπ‘’ ∈ 𝐽 𝑦 = (𝑒 ∩ π‘Œ)))
50 eleq2 2823 . . . . . . . . 9 (𝑦 = (𝑒 ∩ π‘Œ) β†’ (π‘₯ ∈ 𝑦 ↔ π‘₯ ∈ (𝑒 ∩ π‘Œ)))
51 elin 3965 . . . . . . . . . . 11 (π‘₯ ∈ (𝑒 ∩ π‘Œ) ↔ (π‘₯ ∈ 𝑒 ∧ π‘₯ ∈ π‘Œ))
5251rbaib 540 . . . . . . . . . 10 (π‘₯ ∈ π‘Œ β†’ (π‘₯ ∈ (𝑒 ∩ π‘Œ) ↔ π‘₯ ∈ 𝑒))
5352adantl 483 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ (π‘₯ ∈ (𝑒 ∩ π‘Œ) ↔ π‘₯ ∈ 𝑒))
5450, 53sylan9bbr 512 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑦 = (𝑒 ∩ π‘Œ)) β†’ (π‘₯ ∈ 𝑦 ↔ π‘₯ ∈ 𝑒))
55 vex 3479 . . . . . . . . . . . 12 𝑠 ∈ V
5655inex1 5318 . . . . . . . . . . 11 (𝑠 ∩ π‘Œ) ∈ V
5756a1i 11 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑠 ∈ 𝐹) β†’ (𝑠 ∩ π‘Œ) ∈ V)
58 elrest 17373 . . . . . . . . . . . 12 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (𝑧 ∈ (𝐹 β†Ύt π‘Œ) ↔ βˆƒπ‘  ∈ 𝐹 𝑧 = (𝑠 ∩ π‘Œ)))
59583adant1 1131 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (𝑧 ∈ (𝐹 β†Ύt π‘Œ) ↔ βˆƒπ‘  ∈ 𝐹 𝑧 = (𝑠 ∩ π‘Œ)))
6059adantr 482 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ (𝑧 ∈ (𝐹 β†Ύt π‘Œ) ↔ βˆƒπ‘  ∈ 𝐹 𝑧 = (𝑠 ∩ π‘Œ)))
61 ineq2 4207 . . . . . . . . . . . 12 (𝑧 = (𝑠 ∩ π‘Œ) β†’ (𝑦 ∩ 𝑧) = (𝑦 ∩ (𝑠 ∩ π‘Œ)))
6261adantl 483 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 = (𝑠 ∩ π‘Œ)) β†’ (𝑦 ∩ 𝑧) = (𝑦 ∩ (𝑠 ∩ π‘Œ)))
6362neeq1d 3001 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑧 = (𝑠 ∩ π‘Œ)) β†’ ((𝑦 ∩ 𝑧) β‰  βˆ… ↔ (𝑦 ∩ (𝑠 ∩ π‘Œ)) β‰  βˆ…))
6457, 60, 63ralxfr2d 5409 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ (βˆ€π‘§ ∈ (𝐹 β†Ύt π‘Œ)(𝑦 ∩ 𝑧) β‰  βˆ… ↔ βˆ€π‘  ∈ 𝐹 (𝑦 ∩ (𝑠 ∩ π‘Œ)) β‰  βˆ…))
65 ineq1 4206 . . . . . . . . . . . 12 (𝑦 = (𝑒 ∩ π‘Œ) β†’ (𝑦 ∩ (𝑠 ∩ π‘Œ)) = ((𝑒 ∩ π‘Œ) ∩ (𝑠 ∩ π‘Œ)))
66 inindir 4228 . . . . . . . . . . . 12 ((𝑒 ∩ 𝑠) ∩ π‘Œ) = ((𝑒 ∩ π‘Œ) ∩ (𝑠 ∩ π‘Œ))
6765, 66eqtr4di 2791 . . . . . . . . . . 11 (𝑦 = (𝑒 ∩ π‘Œ) β†’ (𝑦 ∩ (𝑠 ∩ π‘Œ)) = ((𝑒 ∩ 𝑠) ∩ π‘Œ))
6867neeq1d 3001 . . . . . . . . . 10 (𝑦 = (𝑒 ∩ π‘Œ) β†’ ((𝑦 ∩ (𝑠 ∩ π‘Œ)) β‰  βˆ… ↔ ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ…))
6968ralbidv 3178 . . . . . . . . 9 (𝑦 = (𝑒 ∩ π‘Œ) β†’ (βˆ€π‘  ∈ 𝐹 (𝑦 ∩ (𝑠 ∩ π‘Œ)) β‰  βˆ… ↔ βˆ€π‘  ∈ 𝐹 ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ…))
7064, 69sylan9bb 511 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑦 = (𝑒 ∩ π‘Œ)) β†’ (βˆ€π‘§ ∈ (𝐹 β†Ύt π‘Œ)(𝑦 ∩ 𝑧) β‰  βˆ… ↔ βˆ€π‘  ∈ 𝐹 ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ…))
7154, 70imbi12d 345 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) ∧ 𝑦 = (𝑒 ∩ π‘Œ)) β†’ ((π‘₯ ∈ 𝑦 β†’ βˆ€π‘§ ∈ (𝐹 β†Ύt π‘Œ)(𝑦 ∩ 𝑧) β‰  βˆ…) ↔ (π‘₯ ∈ 𝑒 β†’ βˆ€π‘  ∈ 𝐹 ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ…)))
7246, 49, 71ralxfr2d 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†’ βˆ€π‘§ ∈ (𝐹 β†Ύt π‘Œ)(𝑦 ∩ 𝑧) β‰  βˆ…) ↔ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆ€π‘  ∈ 𝐹 ((𝑒 ∩ 𝑠) ∩ π‘Œ) β‰  βˆ…)))
731adantr 482 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
7411adantr 482 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
753sselda 3983 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ π‘₯ ∈ 𝑋)
76 fclsopn 23518 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (π‘₯ ∈ (𝐽 fClus 𝐹) ↔ (π‘₯ ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆ€π‘‘ ∈ 𝐹 (𝑒 ∩ 𝑑) β‰  βˆ…))))
7776baibd 541 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯ ∈ (𝐽 fClus 𝐹) ↔ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆ€π‘‘ ∈ 𝐹 (𝑒 ∩ 𝑑) β‰  βˆ…)))
7873, 74, 75, 77syl21anc 837 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ (π‘₯ ∈ (𝐽 fClus 𝐹) ↔ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆ€π‘‘ ∈ 𝐹 (𝑒 ∩ 𝑑) β‰  βˆ…)))
7943, 72, 783bitr4d 311 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) ∧ π‘₯ ∈ π‘Œ) β†’ (βˆ€π‘¦ ∈ (𝐽 β†Ύt π‘Œ)(π‘₯ ∈ 𝑦 β†’ βˆ€π‘§ ∈ (𝐹 β†Ύt π‘Œ)(𝑦 ∩ 𝑧) β‰  βˆ…) ↔ π‘₯ ∈ (𝐽 fClus 𝐹)))
8079pm5.32da 580 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ ((π‘₯ ∈ π‘Œ ∧ βˆ€π‘¦ ∈ (𝐽 β†Ύt π‘Œ)(π‘₯ ∈ 𝑦 β†’ βˆ€π‘§ ∈ (𝐹 β†Ύt π‘Œ)(𝑦 ∩ 𝑧) β‰  βˆ…)) ↔ (π‘₯ ∈ π‘Œ ∧ π‘₯ ∈ (𝐽 fClus 𝐹))))
8116, 80bitrd 279 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (π‘₯ ∈ ((𝐽 β†Ύt π‘Œ) fClus (𝐹 β†Ύt π‘Œ)) ↔ (π‘₯ ∈ π‘Œ ∧ π‘₯ ∈ (𝐽 fClus 𝐹))))
82 elin 3965 . . . 4 (π‘₯ ∈ ((𝐽 fClus 𝐹) ∩ π‘Œ) ↔ (π‘₯ ∈ (𝐽 fClus 𝐹) ∧ π‘₯ ∈ π‘Œ))
8382biancomi 464 . . 3 (π‘₯ ∈ ((𝐽 fClus 𝐹) ∩ π‘Œ) ↔ (π‘₯ ∈ π‘Œ ∧ π‘₯ ∈ (𝐽 fClus 𝐹)))
8481, 83bitr4di 289 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ (π‘₯ ∈ ((𝐽 β†Ύt π‘Œ) fClus (𝐹 β†Ύt π‘Œ)) ↔ π‘₯ ∈ ((𝐽 fClus 𝐹) ∩ π‘Œ)))
8584eqrdv 2731 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘Œ ∈ 𝐹) β†’ ((𝐽 β†Ύt π‘Œ) fClus (𝐹 β†Ύt π‘Œ)) = ((𝐽 fClus 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   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  β€˜cfv 6544  (class class class)co 7409   β†Ύt crest 17366  fBascfbas 20932  TopOnctopon 22412  Filcfil 23349   fClus cfcls 23440
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-iin 5001  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-cld 22523  df-ntr 22524  df-cls 22525  df-fil 23350  df-fcls 23445
This theorem is referenced by:  relcmpcmet  24835
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