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Theorem fclsrest 22548
Description: The set of cluster points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
fclsrest ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝐽t 𝑌) fClus (𝐹t 𝑌)) = ((𝐽 fClus 𝐹) ∩ 𝑌))

Proof of Theorem fclsrest
Dummy variables 𝑠 𝑡 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1130 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2 filelss 22376 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
323adant1 1124 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
4 resttopon 21685 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
51, 3, 4syl2anc 584 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
6 filfbas 22372 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
763ad2ant2 1128 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (fBas‘𝑋))
8 simp3 1132 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝐹)
9 fbncp 22363 . . . . . . 7 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
107, 8, 9syl2anc 584 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
11 simp2 1131 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (Fil‘𝑋))
12 trfil3 22412 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝑋) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1311, 3, 12syl2anc 584 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1410, 13mpbird 258 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐹t 𝑌) ∈ (Fil‘𝑌))
15 fclsopn 22538 . . . . 5 (((𝐽t 𝑌) ∈ (TopOn‘𝑌) ∧ (𝐹t 𝑌) ∈ (Fil‘𝑌)) → (𝑥 ∈ ((𝐽t 𝑌) fClus (𝐹t 𝑌)) ↔ (𝑥𝑌 ∧ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅))))
165, 14, 15syl2anc 584 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fClus (𝐹t 𝑌)) ↔ (𝑥𝑌 ∧ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅))))
17 in32 4202 . . . . . . . . . . . . . 14 ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ 𝑠)
18 ineq2 4187 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → ((𝑢𝑌) ∩ 𝑠) = ((𝑢𝑌) ∩ 𝑡))
1917, 18syl5eq 2873 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ 𝑡))
2019neeq1d 3080 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (((𝑢𝑠) ∩ 𝑌) ≠ ∅ ↔ ((𝑢𝑌) ∩ 𝑡) ≠ ∅))
2120rspccv 3624 . . . . . . . . . . 11 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → (𝑡𝐹 → ((𝑢𝑌) ∩ 𝑡) ≠ ∅))
22 inss1 4209 . . . . . . . . . . . . 13 (𝑢𝑌) ⊆ 𝑢
23 ssrin 4214 . . . . . . . . . . . . 13 ((𝑢𝑌) ⊆ 𝑢 → ((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡))
2422, 23ax-mp 5 . . . . . . . . . . . 12 ((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡)
25 ssn0 4358 . . . . . . . . . . . 12 ((((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡) ∧ ((𝑢𝑌) ∩ 𝑡) ≠ ∅) → (𝑢𝑡) ≠ ∅)
2624, 25mpan 686 . . . . . . . . . . 11 (((𝑢𝑌) ∩ 𝑡) ≠ ∅ → (𝑢𝑡) ≠ ∅)
2721, 26syl6 35 . . . . . . . . . 10 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → (𝑡𝐹 → (𝑢𝑡) ≠ ∅))
2827ralrimiv 3186 . . . . . . . . 9 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)
2911ad3antrrr 726 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → 𝐹 ∈ (Fil‘𝑋))
30 simpr 485 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → 𝑠𝐹)
318ad3antrrr 726 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → 𝑌𝐹)
32 filin 22378 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹𝑌𝐹) → (𝑠𝑌) ∈ 𝐹)
3329, 30, 31, 32syl3anc 1365 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → (𝑠𝑌) ∈ 𝐹)
34 ineq2 4187 . . . . . . . . . . . . . 14 (𝑡 = (𝑠𝑌) → (𝑢𝑡) = (𝑢 ∩ (𝑠𝑌)))
35 inass 4200 . . . . . . . . . . . . . 14 ((𝑢𝑠) ∩ 𝑌) = (𝑢 ∩ (𝑠𝑌))
3634, 35syl6eqr 2879 . . . . . . . . . . . . 13 (𝑡 = (𝑠𝑌) → (𝑢𝑡) = ((𝑢𝑠) ∩ 𝑌))
3736neeq1d 3080 . . . . . . . . . . . 12 (𝑡 = (𝑠𝑌) → ((𝑢𝑡) ≠ ∅ ↔ ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
3837rspcv 3622 . . . . . . . . . . 11 ((𝑠𝑌) ∈ 𝐹 → (∀𝑡𝐹 (𝑢𝑡) ≠ ∅ → ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
3933, 38syl 17 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → (∀𝑡𝐹 (𝑢𝑡) ≠ ∅ → ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
4039ralrimdva 3194 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → (∀𝑡𝐹 (𝑢𝑡) ≠ ∅ → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
4128, 40impbid2 227 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ ↔ ∀𝑡𝐹 (𝑢𝑡) ≠ ∅))
4241imbi2d 342 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → ((𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅) ↔ (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
4342ralbidva 3201 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑢𝐽 (𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
44 vex 3503 . . . . . . . . 9 𝑢 ∈ V
4544inex1 5218 . . . . . . . 8 (𝑢𝑌) ∈ V
4645a1i 11 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → (𝑢𝑌) ∈ V)
47 elrest 16691 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝐹) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑦 = (𝑢𝑌)))
48473adant2 1125 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑦 = (𝑢𝑌)))
4948adantr 481 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑦 = (𝑢𝑌)))
50 eleq2 2906 . . . . . . . . 9 (𝑦 = (𝑢𝑌) → (𝑥𝑦𝑥 ∈ (𝑢𝑌)))
51 elin 4173 . . . . . . . . . . 11 (𝑥 ∈ (𝑢𝑌) ↔ (𝑥𝑢𝑥𝑌))
5251rbaib 539 . . . . . . . . . 10 (𝑥𝑌 → (𝑥 ∈ (𝑢𝑌) ↔ 𝑥𝑢))
5352adantl 482 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝑢𝑌) ↔ 𝑥𝑢))
5450, 53sylan9bbr 511 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → (𝑥𝑦𝑥𝑢))
55 vex 3503 . . . . . . . . . . . 12 𝑠 ∈ V
5655inex1 5218 . . . . . . . . . . 11 (𝑠𝑌) ∈ V
5756a1i 11 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑠𝐹) → (𝑠𝑌) ∈ V)
58 elrest 16691 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑧 ∈ (𝐹t 𝑌) ↔ ∃𝑠𝐹 𝑧 = (𝑠𝑌)))
59583adant1 1124 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑧 ∈ (𝐹t 𝑌) ↔ ∃𝑠𝐹 𝑧 = (𝑠𝑌)))
6059adantr 481 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑧 ∈ (𝐹t 𝑌) ↔ ∃𝑠𝐹 𝑧 = (𝑠𝑌)))
61 ineq2 4187 . . . . . . . . . . . 12 (𝑧 = (𝑠𝑌) → (𝑦𝑧) = (𝑦 ∩ (𝑠𝑌)))
6261adantl 482 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧 = (𝑠𝑌)) → (𝑦𝑧) = (𝑦 ∩ (𝑠𝑌)))
6362neeq1d 3080 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧 = (𝑠𝑌)) → ((𝑦𝑧) ≠ ∅ ↔ (𝑦 ∩ (𝑠𝑌)) ≠ ∅))
6457, 60, 63ralxfr2d 5307 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅ ↔ ∀𝑠𝐹 (𝑦 ∩ (𝑠𝑌)) ≠ ∅))
65 ineq1 4185 . . . . . . . . . . . 12 (𝑦 = (𝑢𝑌) → (𝑦 ∩ (𝑠𝑌)) = ((𝑢𝑌) ∩ (𝑠𝑌)))
66 inindir 4208 . . . . . . . . . . . 12 ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ (𝑠𝑌))
6765, 66syl6eqr 2879 . . . . . . . . . . 11 (𝑦 = (𝑢𝑌) → (𝑦 ∩ (𝑠𝑌)) = ((𝑢𝑠) ∩ 𝑌))
6867neeq1d 3080 . . . . . . . . . 10 (𝑦 = (𝑢𝑌) → ((𝑦 ∩ (𝑠𝑌)) ≠ ∅ ↔ ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
6968ralbidv 3202 . . . . . . . . 9 (𝑦 = (𝑢𝑌) → (∀𝑠𝐹 (𝑦 ∩ (𝑠𝑌)) ≠ ∅ ↔ ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
7064, 69sylan9bb 510 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → (∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅ ↔ ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
7154, 70imbi12d 346 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → ((𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅) ↔ (𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅)))
7246, 49, 71ralxfr2d 5307 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅)))
731adantr 481 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐽 ∈ (TopOn‘𝑋))
7411adantr 481 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐹 ∈ (Fil‘𝑋))
753sselda 3971 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝑥𝑋)
76 fclsopn 22538 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅))))
7776baibd 540 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
7873, 74, 75, 77syl21anc 835 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
7943, 72, 783bitr4d 312 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅) ↔ 𝑥 ∈ (𝐽 fClus 𝐹)))
8079pm5.32da 579 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝑥𝑌 ∧ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅)) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fClus 𝐹))))
8116, 80bitrd 280 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fClus (𝐹t 𝑌)) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fClus 𝐹))))
82 elin 4173 . . . 4 (𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fClus 𝐹) ∧ 𝑥𝑌))
8382biancomi 463 . . 3 (𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fClus 𝐹)))
8481, 83syl6bbr 290 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fClus (𝐹t 𝑌)) ↔ 𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌)))
8584eqrdv 2824 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝐽t 𝑌) fClus (𝐹t 𝑌)) = ((𝐽 fClus 𝐹) ∩ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  Vcvv 3500  cdif 3937  cin 3939  wss 3940  c0 4295  cfv 6352  (class class class)co 7148  t crest 16684  fBascfbas 20449  TopOnctopon 21434  Filcfil 22369   fClus cfcls 22460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-oadd 8097  df-er 8279  df-en 8499  df-fin 8502  df-fi 8864  df-rest 16686  df-topgen 16707  df-fbas 20458  df-fg 20459  df-top 21418  df-topon 21435  df-bases 21470  df-cld 21543  df-ntr 21544  df-cls 21545  df-fil 22370  df-fcls 22465
This theorem is referenced by:  relcmpcmet  23836
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