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Theorem fclsrest 22047
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 1129 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2 filelss 21875 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
323adant1 1123 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
4 resttopon 21185 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
51, 3, 4syl2anc 565 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
6 filfbas 21871 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
763ad2ant2 1127 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (fBas‘𝑋))
8 simp3 1131 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝐹)
9 fbncp 21862 . . . . . . 7 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
107, 8, 9syl2anc 565 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
11 simp2 1130 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (Fil‘𝑋))
12 trfil3 21911 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝑋) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1311, 3, 12syl2anc 565 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1410, 13mpbird 247 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐹t 𝑌) ∈ (Fil‘𝑌))
15 fclsopn 22037 . . . . 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 565 . . . 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 3972 . . . . . . . . . . . . . 14 ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ 𝑠)
18 ineq2 3957 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → ((𝑢𝑌) ∩ 𝑠) = ((𝑢𝑌) ∩ 𝑡))
1917, 18syl5eq 2816 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ 𝑡))
2019neeq1d 3001 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (((𝑢𝑠) ∩ 𝑌) ≠ ∅ ↔ ((𝑢𝑌) ∩ 𝑡) ≠ ∅))
2120rspccv 3455 . . . . . . . . . . 11 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → (𝑡𝐹 → ((𝑢𝑌) ∩ 𝑡) ≠ ∅))
22 inss1 3979 . . . . . . . . . . . . 13 (𝑢𝑌) ⊆ 𝑢
23 ssrin 3984 . . . . . . . . . . . . 13 ((𝑢𝑌) ⊆ 𝑢 → ((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡))
2422, 23ax-mp 5 . . . . . . . . . . . 12 ((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡)
25 ssn0 4118 . . . . . . . . . . . 12 ((((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡) ∧ ((𝑢𝑌) ∩ 𝑡) ≠ ∅) → (𝑢𝑡) ≠ ∅)
2624, 25mpan 662 . . . . . . . . . . 11 (((𝑢𝑌) ∩ 𝑡) ≠ ∅ → (𝑢𝑡) ≠ ∅)
2721, 26syl6 35 . . . . . . . . . 10 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → (𝑡𝐹 → (𝑢𝑡) ≠ ∅))
2827ralrimiv 3113 . . . . . . . . 9 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)
2911ad3antrrr 701 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → 𝐹 ∈ (Fil‘𝑋))
30 simpr 471 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → 𝑠𝐹)
318ad3antrrr 701 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → 𝑌𝐹)
32 filin 21877 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹𝑌𝐹) → (𝑠𝑌) ∈ 𝐹)
3329, 30, 31, 32syl3anc 1475 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → (𝑠𝑌) ∈ 𝐹)
34 ineq2 3957 . . . . . . . . . . . . . 14 (𝑡 = (𝑠𝑌) → (𝑢𝑡) = (𝑢 ∩ (𝑠𝑌)))
35 inass 3970 . . . . . . . . . . . . . 14 ((𝑢𝑠) ∩ 𝑌) = (𝑢 ∩ (𝑠𝑌))
3634, 35syl6eqr 2822 . . . . . . . . . . . . 13 (𝑡 = (𝑠𝑌) → (𝑢𝑡) = ((𝑢𝑠) ∩ 𝑌))
3736neeq1d 3001 . . . . . . . . . . . 12 (𝑡 = (𝑠𝑌) → ((𝑢𝑡) ≠ ∅ ↔ ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
3837rspcv 3454 . . . . . . . . . . 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 3117 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → (∀𝑡𝐹 (𝑢𝑡) ≠ ∅ → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
4128, 40impbid2 216 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ ↔ ∀𝑡𝐹 (𝑢𝑡) ≠ ∅))
4241imbi2d 329 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → ((𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅) ↔ (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
4342ralbidva 3133 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑢𝐽 (𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
44 vex 3352 . . . . . . . . 9 𝑢 ∈ V
4544inex1 4930 . . . . . . . 8 (𝑢𝑌) ∈ V
4645a1i 11 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → (𝑢𝑌) ∈ V)
47 elrest 16295 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝐹) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑦 = (𝑢𝑌)))
48473adant2 1124 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑦 = (𝑢𝑌)))
4948adantr 466 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑦 = (𝑢𝑌)))
50 eleq2 2838 . . . . . . . . 9 (𝑦 = (𝑢𝑌) → (𝑥𝑦𝑥 ∈ (𝑢𝑌)))
51 elin 3945 . . . . . . . . . . 11 (𝑥 ∈ (𝑢𝑌) ↔ (𝑥𝑢𝑥𝑌))
5251rbaib 520 . . . . . . . . . 10 (𝑥𝑌 → (𝑥 ∈ (𝑢𝑌) ↔ 𝑥𝑢))
5352adantl 467 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝑢𝑌) ↔ 𝑥𝑢))
5450, 53sylan9bbr 494 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → (𝑥𝑦𝑥𝑢))
55 vex 3352 . . . . . . . . . . . 12 𝑠 ∈ V
5655inex1 4930 . . . . . . . . . . 11 (𝑠𝑌) ∈ V
5756a1i 11 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑠𝐹) → (𝑠𝑌) ∈ V)
58 elrest 16295 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑧 ∈ (𝐹t 𝑌) ↔ ∃𝑠𝐹 𝑧 = (𝑠𝑌)))
59583adant1 1123 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑧 ∈ (𝐹t 𝑌) ↔ ∃𝑠𝐹 𝑧 = (𝑠𝑌)))
6059adantr 466 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑧 ∈ (𝐹t 𝑌) ↔ ∃𝑠𝐹 𝑧 = (𝑠𝑌)))
61 ineq2 3957 . . . . . . . . . . . 12 (𝑧 = (𝑠𝑌) → (𝑦𝑧) = (𝑦 ∩ (𝑠𝑌)))
6261adantl 467 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧 = (𝑠𝑌)) → (𝑦𝑧) = (𝑦 ∩ (𝑠𝑌)))
6362neeq1d 3001 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧 = (𝑠𝑌)) → ((𝑦𝑧) ≠ ∅ ↔ (𝑦 ∩ (𝑠𝑌)) ≠ ∅))
6457, 60, 63ralxfr2d 5010 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅ ↔ ∀𝑠𝐹 (𝑦 ∩ (𝑠𝑌)) ≠ ∅))
65 ineq1 3956 . . . . . . . . . . . 12 (𝑦 = (𝑢𝑌) → (𝑦 ∩ (𝑠𝑌)) = ((𝑢𝑌) ∩ (𝑠𝑌)))
66 inindir 3978 . . . . . . . . . . . 12 ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ (𝑠𝑌))
6765, 66syl6eqr 2822 . . . . . . . . . . 11 (𝑦 = (𝑢𝑌) → (𝑦 ∩ (𝑠𝑌)) = ((𝑢𝑠) ∩ 𝑌))
6867neeq1d 3001 . . . . . . . . . 10 (𝑦 = (𝑢𝑌) → ((𝑦 ∩ (𝑠𝑌)) ≠ ∅ ↔ ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
6968ralbidv 3134 . . . . . . . . 9 (𝑦 = (𝑢𝑌) → (∀𝑠𝐹 (𝑦 ∩ (𝑠𝑌)) ≠ ∅ ↔ ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
7064, 69sylan9bb 493 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → (∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅ ↔ ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
7154, 70imbi12d 333 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → ((𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅) ↔ (𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅)))
7246, 49, 71ralxfr2d 5010 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅)))
731adantr 466 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐽 ∈ (TopOn‘𝑋))
7411adantr 466 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐹 ∈ (Fil‘𝑋))
753sselda 3750 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝑥𝑋)
76 fclsopn 22037 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅))))
7776baibd 521 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
7873, 74, 75, 77syl21anc 1474 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
7943, 72, 783bitr4d 300 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅) ↔ 𝑥 ∈ (𝐽 fClus 𝐹)))
8079pm5.32da 560 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝑥𝑌 ∧ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅)) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fClus 𝐹))))
8116, 80bitrd 268 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fClus (𝐹t 𝑌)) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fClus 𝐹))))
82 elin 3945 . . . 4 (𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fClus 𝐹) ∧ 𝑥𝑌))
83 ancom 452 . . . 4 ((𝑥 ∈ (𝐽 fClus 𝐹) ∧ 𝑥𝑌) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fClus 𝐹)))
8482, 83bitri 264 . . 3 (𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fClus 𝐹)))
8581, 84syl6bbr 278 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fClus (𝐹t 𝑌)) ↔ 𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌)))
8685eqrdv 2768 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 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wral 3060  wrex 3061  Vcvv 3349  cdif 3718  cin 3720  wss 3721  c0 4061  cfv 6031  (class class class)co 6792  t crest 16288  fBascfbas 19948  TopOnctopon 20934  Filcfil 21868   fClus cfcls 21959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-oadd 7716  df-er 7895  df-en 8109  df-fin 8112  df-fi 8472  df-rest 16290  df-topgen 16311  df-fbas 19957  df-fg 19958  df-top 20918  df-topon 20935  df-bases 20970  df-cld 21043  df-ntr 21044  df-cls 21045  df-fil 21869  df-fcls 21964
This theorem is referenced by:  relcmpcmet  23333
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