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Theorem fclsrest 21751
 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 1059 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2 filelss 21579 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
323adant1 1077 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝑋)
4 resttopon 20888 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
51, 3, 4syl2anc 692 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
6 filfbas 21575 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
763ad2ant2 1081 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (fBas‘𝑋))
8 simp3 1061 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝑌𝐹)
9 fbncp 21566 . . . . . . 7 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
107, 8, 9syl2anc 692 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ¬ (𝑋𝑌) ∈ 𝐹)
11 simp2 1060 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → 𝐹 ∈ (Fil‘𝑋))
12 trfil3 21615 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝑋) → ((𝐹t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋𝑌) ∈ 𝐹))
1311, 3, 12syl2anc 692 . . . . . 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 21741 . . . . 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 692 . . . 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 3808 . . . . . . . . . . . . . 14 ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ 𝑠)
18 ineq2 3791 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → ((𝑢𝑌) ∩ 𝑠) = ((𝑢𝑌) ∩ 𝑡))
1917, 18syl5eq 2667 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ 𝑡))
2019neeq1d 2849 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (((𝑢𝑠) ∩ 𝑌) ≠ ∅ ↔ ((𝑢𝑌) ∩ 𝑡) ≠ ∅))
2120rspccv 3295 . . . . . . . . . . 11 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → (𝑡𝐹 → ((𝑢𝑌) ∩ 𝑡) ≠ ∅))
22 inss1 3816 . . . . . . . . . . . . 13 (𝑢𝑌) ⊆ 𝑢
23 ssrin 3821 . . . . . . . . . . . . 13 ((𝑢𝑌) ⊆ 𝑢 → ((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡))
2422, 23ax-mp 5 . . . . . . . . . . . 12 ((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡)
25 ssn0 3953 . . . . . . . . . . . 12 ((((𝑢𝑌) ∩ 𝑡) ⊆ (𝑢𝑡) ∧ ((𝑢𝑌) ∩ 𝑡) ≠ ∅) → (𝑢𝑡) ≠ ∅)
2624, 25mpan 705 . . . . . . . . . . 11 (((𝑢𝑌) ∩ 𝑡) ≠ ∅ → (𝑢𝑡) ≠ ∅)
2721, 26syl6 35 . . . . . . . . . 10 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → (𝑡𝐹 → (𝑢𝑡) ≠ ∅))
2827ralrimiv 2960 . . . . . . . . 9 (∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅ → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)
2911ad3antrrr 765 . . . . . . . . . . . 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 765 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → 𝑌𝐹)
32 filin 21581 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹𝑌𝐹) → (𝑠𝑌) ∈ 𝐹)
3329, 30, 31, 32syl3anc 1323 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) ∧ 𝑠𝐹) → (𝑠𝑌) ∈ 𝐹)
34 ineq2 3791 . . . . . . . . . . . . . 14 (𝑡 = (𝑠𝑌) → (𝑢𝑡) = (𝑢 ∩ (𝑠𝑌)))
35 inass 3806 . . . . . . . . . . . . . 14 ((𝑢𝑠) ∩ 𝑌) = (𝑢 ∩ (𝑠𝑌))
3634, 35syl6eqr 2673 . . . . . . . . . . . . 13 (𝑡 = (𝑠𝑌) → (𝑢𝑡) = ((𝑢𝑠) ∩ 𝑌))
3736neeq1d 2849 . . . . . . . . . . . 12 (𝑡 = (𝑠𝑌) → ((𝑢𝑡) ≠ ∅ ↔ ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
3837rspcv 3294 . . . . . . . . . . 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 2964 . . . . . . . . 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 330 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → ((𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅) ↔ (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
4342ralbidva 2980 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑢𝐽 (𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
44 vex 3192 . . . . . . . . 9 𝑢 ∈ V
4544inex1 4764 . . . . . . . 8 (𝑢𝑌) ∈ V
4645a1i 11 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑢𝐽) → (𝑢𝑌) ∈ V)
47 elrest 16020 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝐹) → (𝑦 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑦 = (𝑢𝑌)))
48473adant2 1078 . . . . . . . 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 2687 . . . . . . . . 9 (𝑦 = (𝑢𝑌) → (𝑥𝑦𝑥 ∈ (𝑢𝑌)))
51 elin 3779 . . . . . . . . . . 11 (𝑥 ∈ (𝑢𝑌) ↔ (𝑥𝑢𝑥𝑌))
5251rbaib 946 . . . . . . . . . 10 (𝑥𝑌 → (𝑥 ∈ (𝑢𝑌) ↔ 𝑥𝑢))
5352adantl 482 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (𝑥 ∈ (𝑢𝑌) ↔ 𝑥𝑢))
5450, 53sylan9bbr 736 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → (𝑥𝑦𝑥𝑢))
55 vex 3192 . . . . . . . . . . . 12 𝑠 ∈ V
5655inex1 4764 . . . . . . . . . . 11 (𝑠𝑌) ∈ V
5756a1i 11 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑠𝐹) → (𝑠𝑌) ∈ V)
58 elrest 16020 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝑧 ∈ (𝐹t 𝑌) ↔ ∃𝑠𝐹 𝑧 = (𝑠𝑌)))
59583adant1 1077 . . . . . . . . . . 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 3791 . . . . . . . . . . . 12 (𝑧 = (𝑠𝑌) → (𝑦𝑧) = (𝑦 ∩ (𝑠𝑌)))
6261adantl 482 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧 = (𝑠𝑌)) → (𝑦𝑧) = (𝑦 ∩ (𝑠𝑌)))
6362neeq1d 2849 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑧 = (𝑠𝑌)) → ((𝑦𝑧) ≠ ∅ ↔ (𝑦 ∩ (𝑠𝑌)) ≠ ∅))
6457, 60, 63ralxfr2d 4847 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → (∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅ ↔ ∀𝑠𝐹 (𝑦 ∩ (𝑠𝑌)) ≠ ∅))
65 ineq1 3790 . . . . . . . . . . . 12 (𝑦 = (𝑢𝑌) → (𝑦 ∩ (𝑠𝑌)) = ((𝑢𝑌) ∩ (𝑠𝑌)))
66 inindir 3814 . . . . . . . . . . . 12 ((𝑢𝑠) ∩ 𝑌) = ((𝑢𝑌) ∩ (𝑠𝑌))
6765, 66syl6eqr 2673 . . . . . . . . . . 11 (𝑦 = (𝑢𝑌) → (𝑦 ∩ (𝑠𝑌)) = ((𝑢𝑠) ∩ 𝑌))
6867neeq1d 2849 . . . . . . . . . 10 (𝑦 = (𝑢𝑌) → ((𝑦 ∩ (𝑠𝑌)) ≠ ∅ ↔ ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
6968ralbidv 2981 . . . . . . . . 9 (𝑦 = (𝑢𝑌) → (∀𝑠𝐹 (𝑦 ∩ (𝑠𝑌)) ≠ ∅ ↔ ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
7064, 69sylan9bb 735 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → (∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅ ↔ ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅))
7154, 70imbi12d 334 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) ∧ 𝑦 = (𝑢𝑌)) → ((𝑥𝑦 → ∀𝑧 ∈ (𝐹t 𝑌)(𝑦𝑧) ≠ ∅) ↔ (𝑥𝑢 → ∀𝑠𝐹 ((𝑢𝑠) ∩ 𝑌) ≠ ∅)))
7246, 49, 71ralxfr2d 4847 . . . . . 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 3587 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) ∧ 𝑥𝑌) → 𝑥𝑋)
76 fclsopn 21741 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅))))
7776baibd 947 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑢𝐽 (𝑥𝑢 → ∀𝑡𝐹 (𝑢𝑡) ≠ ∅)))
7873, 74, 75, 77syl21anc 1322 . . . . . 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 672 . . . 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 3779 . . . 4 (𝑥 ∈ ((𝐽 fClus 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fClus 𝐹) ∧ 𝑥𝑌))
83 ancom 466 . . . 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 2619 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 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908  Vcvv 3189   ∖ cdif 3556   ∩ cin 3558   ⊆ wss 3559  ∅c0 3896  ‘cfv 5852  (class class class)co 6610   ↾t crest 16013  fBascfbas 19666  TopOnctopon 20647  Filcfil 21572   fClus cfcls 21663 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-oadd 7516  df-er 7694  df-en 7908  df-fin 7911  df-fi 8269  df-rest 16015  df-topgen 16036  df-fbas 19675  df-fg 19676  df-top 20631  df-topon 20648  df-bases 20674  df-cld 20746  df-ntr 20747  df-cls 20748  df-fil 21573  df-fcls 21668 This theorem is referenced by:  relcmpcmet  23038
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