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Theorem fclsrest 22729
 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 1133 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2 filelss 22557 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
323adant1 1127 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
4 resttopon 21866 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
51, 3, 4syl2anc 587 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
6 filfbas 22553 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
763ad2ant2 1131 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (fBas‘𝑋))
8 simp3 1135 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝐹)
9 fbncp 22544 . . . . . . 7 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
107, 8, 9syl2anc 587 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
11 simp2 1134 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (Fil‘𝑋))
12 trfil3 22593 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝑋) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1311, 3, 12syl2anc 587 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1410, 13mpbird 260 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐹t 𝑌) ∈ (Fil‘𝑌))
15 fclsopn 22719 . . . . 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 587 . . . 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 4128 . . . . . . . . . . . . . 14 ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ 𝑠)
18 ineq2 4113 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → ((𝑢𝑌) ∩ 𝑠) = ((𝑢𝑌) ∩ 𝑡))
1917, 18syl5eq 2805 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ 𝑡))
2019neeq1d 3010 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (((𝑢𝑠) ∩ 𝑌) ≠ ∅ ↔ ((𝑢𝑌) ∩ 𝑡) ≠ ∅))
2120rspccv 3540 . . . . . . . . . . 11 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → (𝑡𝐹 → ((𝑢𝑌) ∩ 𝑡) ≠ ∅))
22 inss1 4135 . . . . . . . . . . . . 13 (𝑢𝑌) ⊆ 𝑢
23 ssrin 4140 . . . . . . . . . . . . 13 ((𝑢𝑌) ⊆ 𝑢 → ((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡))
2422, 23ax-mp 5 . . . . . . . . . . . 12 ((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡)
25 ssn0 4299 . . . . . . . . . . . 12 ((((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡) ∧ ((𝑢𝑌) ∩ 𝑡) ≠ ∅) → (𝑢𝑡) ≠ ∅)
2624, 25mpan 689 . . . . . . . . . . 11 (((𝑢𝑌) ∩ 𝑡) ≠ ∅ → (𝑢𝑡) ≠ ∅)
2721, 26syl6 35 . . . . . . . . . 10 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → (𝑡𝐹 → (𝑢𝑡) ≠ ∅))
2827ralrimiv 3112 . . . . . . . . 9 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)
2911ad3antrrr 729 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → 𝐹 ∈ (Fil‘𝑋))
30 simpr 488 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → 𝑠𝐹)
318ad3antrrr 729 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → 𝑌𝐹)
32 filin 22559 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹𝑌𝐹) → (𝑠𝑌) ∈ 𝐹)
3329, 30, 31, 32syl3anc 1368 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → (𝑠𝑌) ∈ 𝐹)
34 ineq2 4113 . . . . . . . . . . . . . 14 (𝑡 = (𝑠𝑌) → (𝑢𝑡) = (𝑢 ∩ (𝑠𝑌)))
35 inass 4126 . . . . . . . . . . . . . 14 ((𝑢𝑠) ∩ 𝑌) = (𝑢 ∩ (𝑠𝑌))
3634, 35eqtr4di 2811 . . . . . . . . . . . . 13 (𝑡 = (𝑠𝑌) → (𝑢𝑡) = ((𝑢𝑠) ∩ 𝑌))
3736neeq1d 3010 . . . . . . . . . . . 12 (𝑡 = (𝑠𝑌) → ((𝑢𝑡) ≠ ∅ ↔ ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
3837rspcv 3538 . . . . . . . . . . 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 3118 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → (∀𝑡𝐹 (𝑢𝑡) ≠ ∅ → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
4128, 40impbid2 229 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ ↔ ∀𝑡𝐹 (𝑢𝑡) ≠ ∅))
4241imbi2d 344 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → ((𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅) ↔ (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
4342ralbidva 3125 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑢𝐽 (𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
44 vex 3413 . . . . . . . . 9 𝑢 ∈ V
4544inex1 5190 . . . . . . . 8 (𝑢𝑌) ∈ V
4645a1i 11 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → (𝑢𝑌) ∈ V)
47 elrest 16764 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝐹) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑦 = (𝑢𝑌)))
48473adant2 1128 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑦 = (𝑢𝑌)))
4948adantr 484 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑦 = (𝑢𝑌)))
50 eleq2 2840 . . . . . . . . 9 (𝑦 = (𝑢𝑌) → (𝑥𝑦𝑥 ∈ (𝑢𝑌)))
51 elin 3876 . . . . . . . . . . 11 (𝑥 ∈ (𝑢𝑌) ↔ (𝑥𝑢𝑥𝑌))
5251rbaib 542 . . . . . . . . . 10 (𝑥𝑌 → (𝑥 ∈ (𝑢𝑌) ↔ 𝑥𝑢))
5352adantl 485 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝑢𝑌) ↔ 𝑥𝑢))
5450, 53sylan9bbr 514 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → (𝑥𝑦𝑥𝑢))
55 vex 3413 . . . . . . . . . . . 12 𝑠 ∈ V
5655inex1 5190 . . . . . . . . . . 11 (𝑠𝑌) ∈ V
5756a1i 11 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑠𝐹) → (𝑠𝑌) ∈ V)
58 elrest 16764 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑧 ∈ (𝐹t 𝑌) ↔ ∃𝑠𝐹 𝑧 = (𝑠𝑌)))
59583adant1 1127 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑧 ∈ (𝐹t 𝑌) ↔ ∃𝑠𝐹 𝑧 = (𝑠𝑌)))
6059adantr 484 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑧 ∈ (𝐹t 𝑌) ↔ ∃𝑠𝐹 𝑧 = (𝑠𝑌)))
61 ineq2 4113 . . . . . . . . . . . 12 (𝑧 = (𝑠𝑌) → (𝑦𝑧) = (𝑦 ∩ (𝑠𝑌)))
6261adantl 485 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧 = (𝑠𝑌)) → (𝑦𝑧) = (𝑦 ∩ (𝑠𝑌)))
6362neeq1d 3010 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧 = (𝑠𝑌)) → ((𝑦𝑧) ≠ ∅ ↔ (𝑦 ∩ (𝑠𝑌)) ≠ ∅))
6457, 60, 63ralxfr2d 5282 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅ ↔ ∀𝑠𝐹 (𝑦 ∩ (𝑠𝑌)) ≠ ∅))
65 ineq1 4111 . . . . . . . . . . . 12 (𝑦 = (𝑢𝑌) → (𝑦 ∩ (𝑠𝑌)) = ((𝑢𝑌) ∩ (𝑠𝑌)))
66 inindir 4134 . . . . . . . . . . . 12 ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ (𝑠𝑌))
6765, 66eqtr4di 2811 . . . . . . . . . . 11 (𝑦 = (𝑢𝑌) → (𝑦 ∩ (𝑠𝑌)) = ((𝑢𝑠) ∩ 𝑌))
6867neeq1d 3010 . . . . . . . . . 10 (𝑦 = (𝑢𝑌) → ((𝑦 ∩ (𝑠𝑌)) ≠ ∅ ↔ ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
6968ralbidv 3126 . . . . . . . . 9 (𝑦 = (𝑢𝑌) → (∀𝑠𝐹 (𝑦 ∩ (𝑠𝑌)) ≠ ∅ ↔ ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
7064, 69sylan9bb 513 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → (∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅ ↔ ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
7154, 70imbi12d 348 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → ((𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅) ↔ (𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅)))
7246, 49, 71ralxfr2d 5282 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅)))
731adantr 484 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐽 ∈ (TopOn‘𝑋))
7411adantr 484 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐹 ∈ (Fil‘𝑋))
753sselda 3894 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝑥𝑋)
76 fclsopn 22719 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅))))
7776baibd 543 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
7873, 74, 75, 77syl21anc 836 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
7943, 72, 783bitr4d 314 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅) ↔ 𝑥 ∈ (𝐽 fClus 𝐹)))
8079pm5.32da 582 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝑥𝑌 ∧ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅)) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fClus 𝐹))))
8116, 80bitrd 282 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fClus (𝐹t 𝑌)) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fClus 𝐹))))
82 elin 3876 . . . 4 (𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fClus 𝐹) ∧ 𝑥𝑌))
8382biancomi 466 . . 3 (𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fClus 𝐹)))
8481, 83bitr4di 292 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fClus (𝐹t 𝑌)) ↔ 𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌)))
8584eqrdv 2756 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ∖ cdif 3857   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  ‘cfv 6339  (class class class)co 7155   ↾t crest 16757  fBascfbas 20159  TopOnctopon 21615  Filcfil 22550   fClus cfcls 22641 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-en 8533  df-fin 8536  df-fi 8913  df-rest 16759  df-topgen 16780  df-fbas 20168  df-fg 20169  df-top 21599  df-topon 21616  df-bases 21651  df-cld 21724  df-ntr 21725  df-cls 21726  df-fil 22551  df-fcls 22646 This theorem is referenced by:  relcmpcmet  24023
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