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Theorem fclsrest 22107
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 1166 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2 filelss 21935 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
323adant1 1160 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
4 resttopon 21245 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
51, 3, 4syl2anc 579 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
6 filfbas 21931 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
763ad2ant2 1164 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (fBas‘𝑋))
8 simp3 1168 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝐹)
9 fbncp 21922 . . . . . . 7 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
107, 8, 9syl2anc 579 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
11 simp2 1167 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (Fil‘𝑋))
12 trfil3 21971 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝑋) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1311, 3, 12syl2anc 579 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1410, 13mpbird 248 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐹t 𝑌) ∈ (Fil‘𝑌))
15 fclsopn 22097 . . . . 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 579 . . . 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 3985 . . . . . . . . . . . . . 14 ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ 𝑠)
18 ineq2 3970 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → ((𝑢𝑌) ∩ 𝑠) = ((𝑢𝑌) ∩ 𝑡))
1917, 18syl5eq 2811 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ 𝑡))
2019neeq1d 2996 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (((𝑢𝑠) ∩ 𝑌) ≠ ∅ ↔ ((𝑢𝑌) ∩ 𝑡) ≠ ∅))
2120rspccv 3458 . . . . . . . . . . 11 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → (𝑡𝐹 → ((𝑢𝑌) ∩ 𝑡) ≠ ∅))
22 inss1 3992 . . . . . . . . . . . . 13 (𝑢𝑌) ⊆ 𝑢
23 ssrin 3997 . . . . . . . . . . . . 13 ((𝑢𝑌) ⊆ 𝑢 → ((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡))
2422, 23ax-mp 5 . . . . . . . . . . . 12 ((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡)
25 ssn0 4138 . . . . . . . . . . . 12 ((((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡) ∧ ((𝑢𝑌) ∩ 𝑡) ≠ ∅) → (𝑢𝑡) ≠ ∅)
2624, 25mpan 681 . . . . . . . . . . 11 (((𝑢𝑌) ∩ 𝑡) ≠ ∅ → (𝑢𝑡) ≠ ∅)
2721, 26syl6 35 . . . . . . . . . 10 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → (𝑡𝐹 → (𝑢𝑡) ≠ ∅))
2827ralrimiv 3112 . . . . . . . . 9 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)
2911ad3antrrr 721 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → 𝐹 ∈ (Fil‘𝑋))
30 simpr 477 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → 𝑠𝐹)
318ad3antrrr 721 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → 𝑌𝐹)
32 filin 21937 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹𝑌𝐹) → (𝑠𝑌) ∈ 𝐹)
3329, 30, 31, 32syl3anc 1490 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → (𝑠𝑌) ∈ 𝐹)
34 ineq2 3970 . . . . . . . . . . . . . 14 (𝑡 = (𝑠𝑌) → (𝑢𝑡) = (𝑢 ∩ (𝑠𝑌)))
35 inass 3983 . . . . . . . . . . . . . 14 ((𝑢𝑠) ∩ 𝑌) = (𝑢 ∩ (𝑠𝑌))
3634, 35syl6eqr 2817 . . . . . . . . . . . . 13 (𝑡 = (𝑠𝑌) → (𝑢𝑡) = ((𝑢𝑠) ∩ 𝑌))
3736neeq1d 2996 . . . . . . . . . . . 12 (𝑡 = (𝑠𝑌) → ((𝑢𝑡) ≠ ∅ ↔ ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
3837rspcv 3457 . . . . . . . . . . 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 3116 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → (∀𝑡𝐹 (𝑢𝑡) ≠ ∅ → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
4128, 40impbid2 217 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ ↔ ∀𝑡𝐹 (𝑢𝑡) ≠ ∅))
4241imbi2d 331 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → ((𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅) ↔ (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
4342ralbidva 3132 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑢𝐽 (𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
44 vex 3353 . . . . . . . . 9 𝑢 ∈ V
4544inex1 4960 . . . . . . . 8 (𝑢𝑌) ∈ V
4645a1i 11 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → (𝑢𝑌) ∈ V)
47 elrest 16354 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝐹) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑦 = (𝑢𝑌)))
48473adant2 1161 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑦 = (𝑢𝑌)))
4948adantr 472 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑦 = (𝑢𝑌)))
50 eleq2 2833 . . . . . . . . 9 (𝑦 = (𝑢𝑌) → (𝑥𝑦𝑥 ∈ (𝑢𝑌)))
51 elin 3958 . . . . . . . . . . 11 (𝑥 ∈ (𝑢𝑌) ↔ (𝑥𝑢𝑥𝑌))
5251rbaib 534 . . . . . . . . . 10 (𝑥𝑌 → (𝑥 ∈ (𝑢𝑌) ↔ 𝑥𝑢))
5352adantl 473 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝑢𝑌) ↔ 𝑥𝑢))
5450, 53sylan9bbr 506 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → (𝑥𝑦𝑥𝑢))
55 vex 3353 . . . . . . . . . . . 12 𝑠 ∈ V
5655inex1 4960 . . . . . . . . . . 11 (𝑠𝑌) ∈ V
5756a1i 11 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑠𝐹) → (𝑠𝑌) ∈ V)
58 elrest 16354 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑧 ∈ (𝐹t 𝑌) ↔ ∃𝑠𝐹 𝑧 = (𝑠𝑌)))
59583adant1 1160 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑧 ∈ (𝐹t 𝑌) ↔ ∃𝑠𝐹 𝑧 = (𝑠𝑌)))
6059adantr 472 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑧 ∈ (𝐹t 𝑌) ↔ ∃𝑠𝐹 𝑧 = (𝑠𝑌)))
61 ineq2 3970 . . . . . . . . . . . 12 (𝑧 = (𝑠𝑌) → (𝑦𝑧) = (𝑦 ∩ (𝑠𝑌)))
6261adantl 473 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧 = (𝑠𝑌)) → (𝑦𝑧) = (𝑦 ∩ (𝑠𝑌)))
6362neeq1d 2996 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧 = (𝑠𝑌)) → ((𝑦𝑧) ≠ ∅ ↔ (𝑦 ∩ (𝑠𝑌)) ≠ ∅))
6457, 60, 63ralxfr2d 5045 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅ ↔ ∀𝑠𝐹 (𝑦 ∩ (𝑠𝑌)) ≠ ∅))
65 ineq1 3969 . . . . . . . . . . . 12 (𝑦 = (𝑢𝑌) → (𝑦 ∩ (𝑠𝑌)) = ((𝑢𝑌) ∩ (𝑠𝑌)))
66 inindir 3991 . . . . . . . . . . . 12 ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ (𝑠𝑌))
6765, 66syl6eqr 2817 . . . . . . . . . . 11 (𝑦 = (𝑢𝑌) → (𝑦 ∩ (𝑠𝑌)) = ((𝑢𝑠) ∩ 𝑌))
6867neeq1d 2996 . . . . . . . . . 10 (𝑦 = (𝑢𝑌) → ((𝑦 ∩ (𝑠𝑌)) ≠ ∅ ↔ ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
6968ralbidv 3133 . . . . . . . . 9 (𝑦 = (𝑢𝑌) → (∀𝑠𝐹 (𝑦 ∩ (𝑠𝑌)) ≠ ∅ ↔ ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
7064, 69sylan9bb 505 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → (∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅ ↔ ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
7154, 70imbi12d 335 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → ((𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅) ↔ (𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅)))
7246, 49, 71ralxfr2d 5045 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅)))
731adantr 472 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐽 ∈ (TopOn‘𝑋))
7411adantr 472 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝐹 ∈ (Fil‘𝑋))
753sselda 3761 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝑥𝑋)
76 fclsopn 22097 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅))))
7776baibd 535 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
7873, 74, 75, 77syl21anc 866 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
7943, 72, 783bitr4d 302 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅) ↔ 𝑥 ∈ (𝐽 fClus 𝐹)))
8079pm5.32da 574 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝑥𝑌 ∧ ∀𝑦 ∈ (𝐽t 𝑌)(𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅)) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fClus 𝐹))))
8116, 80bitrd 270 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fClus (𝐹t 𝑌)) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fClus 𝐹))))
82 elin 3958 . . . 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 266 . . 3 (𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌) ↔ (𝑥𝑌𝑥 ∈ (𝐽 fClus 𝐹)))
8581, 84syl6bbr 280 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑥 ∈ ((𝐽t 𝑌) fClus (𝐹t 𝑌)) ↔ 𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌)))
8685eqrdv 2763 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  Vcvv 3350  cdif 3729  cin 3731  wss 3732  c0 4079  cfv 6068  (class class class)co 6842  t crest 16347  fBascfbas 20007  TopOnctopon 20994  Filcfil 21928   fClus cfcls 22019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-oadd 7768  df-er 7947  df-en 8161  df-fin 8164  df-fi 8524  df-rest 16349  df-topgen 16370  df-fbas 20016  df-fg 20017  df-top 20978  df-topon 20995  df-bases 21030  df-cld 21103  df-ntr 21104  df-cls 21105  df-fil 21929  df-fcls 22024
This theorem is referenced by:  relcmpcmet  23395
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