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Theorem fclscf 24033
Description: Characterization of fineness of topologies in terms of cluster points. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclscf ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (𝐽𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)))
Distinct variable groups:   𝑓,𝐽   𝑓,𝐾   𝑓,𝑋

Proof of Theorem fclscf
Dummy variables 𝑛 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 767 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐽 ∈ (TopOn‘𝑋))
2 simplr 769 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐾 ∈ (TopOn‘𝑋))
3 fclstopon 24020 . . . . . . . . 9 (𝑥 ∈ (𝐾 fClus 𝑓) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝑓 ∈ (Fil‘𝑋)))
43ad2antll 729 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝑓 ∈ (Fil‘𝑋)))
52, 4mpbid 232 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑓 ∈ (Fil‘𝑋))
6 simprl 771 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐽𝐾)
7 fclsss1 24030 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
81, 5, 6, 7syl3anc 1373 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
9 simprr 773 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑥 ∈ (𝐾 fClus 𝑓))
108, 9sseldd 3984 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑥 ∈ (𝐽 fClus 𝑓))
1110expr 456 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) → (𝑥 ∈ (𝐾 fClus 𝑓) → 𝑥 ∈ (𝐽 fClus 𝑓)))
1211ssrdv 3989 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
1312ralrimivw 3150 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) → ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
14 simpllr 776 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → 𝐾 ∈ (TopOn‘𝑋))
15 toponmax 22932 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑋) → 𝑋𝐾)
16 ssid 4006 . . . . . . . . . . 11 𝑋𝑋
17 eleq2 2830 . . . . . . . . . . . . 13 (𝑢 = 𝑋 → (𝑦𝑢𝑦𝑋))
18 sseq1 4009 . . . . . . . . . . . . 13 (𝑢 = 𝑋 → (𝑢𝑋𝑋𝑋))
1917, 18anbi12d 632 . . . . . . . . . . . 12 (𝑢 = 𝑋 → ((𝑦𝑢𝑢𝑋) ↔ (𝑦𝑋𝑋𝑋)))
2019rspcev 3622 . . . . . . . . . . 11 ((𝑋𝐾 ∧ (𝑦𝑋𝑋𝑋)) → ∃𝑢𝐾 (𝑦𝑢𝑢𝑋))
2116, 20mpanr2 704 . . . . . . . . . 10 ((𝑋𝐾𝑦𝑋) → ∃𝑢𝐾 (𝑦𝑢𝑢𝑋))
2221ex 412 . . . . . . . . 9 (𝑋𝐾 → (𝑦𝑋 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑋)))
2314, 15, 223syl 18 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → (𝑦𝑋 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑋)))
24 eleq2 2830 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑦𝑥𝑦𝑋))
25 sseq2 4010 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑢𝑥𝑢𝑋))
2625anbi2d 630 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝑦𝑢𝑢𝑥) ↔ (𝑦𝑢𝑢𝑋)))
2726rexbidv 3179 . . . . . . . . 9 (𝑥 = 𝑋 → (∃𝑢𝐾 (𝑦𝑢𝑢𝑥) ↔ ∃𝑢𝐾 (𝑦𝑢𝑢𝑋)))
2824, 27imbi12d 344 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑦𝑥 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥)) ↔ (𝑦𝑋 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑋))))
2923, 28syl5ibrcom 247 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → (𝑥 = 𝑋 → (𝑦𝑥 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥))))
30 simplll 775 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝐽 ∈ (TopOn‘𝑋))
31 simprl 771 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑥𝐽)
32 simprrr 782 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑦𝑥)
33 supnfcls 24028 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽𝑦𝑥) → ¬ 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}))
3430, 31, 32, 33syl3anc 1373 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → ¬ 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}))
35 toponss 22933 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
3630, 31, 35syl2anc 584 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑥𝑋)
3736, 32sseldd 3984 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑦𝑋)
38 simpllr 776 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝐾 ∈ (TopOn‘𝑋))
39 toponmax 22932 . . . . . . . . . . . . . . . 16 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
4030, 39syl 17 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑋𝐽)
41 difssd 4137 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → (𝑋𝑥) ⊆ 𝑋)
42 simprrl 781 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑥𝑋)
43 pssdifn0 4368 . . . . . . . . . . . . . . . 16 ((𝑥𝑋𝑥𝑋) → (𝑋𝑥) ≠ ∅)
4436, 42, 43syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → (𝑋𝑥) ≠ ∅)
45 supfil 23903 . . . . . . . . . . . . . . 15 ((𝑋𝐽 ∧ (𝑋𝑥) ⊆ 𝑋 ∧ (𝑋𝑥) ≠ ∅) → {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} ∈ (Fil‘𝑋))
4640, 41, 44, 45syl3anc 1373 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} ∈ (Fil‘𝑋))
47 fclsopn 24022 . . . . . . . . . . . . . 14 ((𝐾 ∈ (TopOn‘𝑋) ∧ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} ∈ (Fil‘𝑋)) → (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}) ↔ (𝑦𝑋 ∧ ∀𝑢𝐾 (𝑦𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅))))
4838, 46, 47syl2anc 584 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}) ↔ (𝑦𝑋 ∧ ∀𝑢𝐾 (𝑦𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅))))
4937, 48mpbirand 707 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}) ↔ ∀𝑢𝐾 (𝑦𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅)))
50 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} → (𝐾 fClus 𝑓) = (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}))
51 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} → (𝐽 fClus 𝑓) = (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}))
5250, 51sseq12d 4017 . . . . . . . . . . . . . 14 (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} → ((𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓) ↔ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}) ⊆ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦})))
53 simplr 769 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
5452, 53, 46rspcdva 3623 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}) ⊆ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}))
5554sseld 3982 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}) → 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦})))
5649, 55sylbird 260 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → (∀𝑢𝐾 (𝑦𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅) → 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦})))
5734, 56mtod 198 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → ¬ ∀𝑢𝐾 (𝑦𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅))
58 rexanali 3102 . . . . . . . . . . 11 (∃𝑢𝐾 (𝑦𝑢 ∧ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅) ↔ ¬ ∀𝑢𝐾 (𝑦𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅))
59 rexnal 3100 . . . . . . . . . . . . . 14 (∃𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} ¬ (𝑢𝑛) ≠ ∅ ↔ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅)
60 sseq2 4010 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑛 → ((𝑋𝑥) ⊆ 𝑦 ↔ (𝑋𝑥) ⊆ 𝑛))
6160elrab 3692 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} ↔ (𝑛 ∈ 𝒫 𝑋 ∧ (𝑋𝑥) ⊆ 𝑛))
62 sslin 4243 . . . . . . . . . . . . . . . . 17 ((𝑋𝑥) ⊆ 𝑛 → (𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛))
6361, 62simplbiim 504 . . . . . . . . . . . . . . . 16 (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} → (𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛))
64 ssn0 4404 . . . . . . . . . . . . . . . . . . . 20 (((𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛) ∧ (𝑢 ∩ (𝑋𝑥)) ≠ ∅) → (𝑢𝑛) ≠ ∅)
6564ex 412 . . . . . . . . . . . . . . . . . . 19 ((𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛) → ((𝑢 ∩ (𝑋𝑥)) ≠ ∅ → (𝑢𝑛) ≠ ∅))
6665necon1bd 2958 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛) → (¬ (𝑢𝑛) ≠ ∅ → (𝑢 ∩ (𝑋𝑥)) = ∅))
67 inssdif0 4374 . . . . . . . . . . . . . . . . . 18 ((𝑢𝑋) ⊆ 𝑥 ↔ (𝑢 ∩ (𝑋𝑥)) = ∅)
6866, 67imbitrrdi 252 . . . . . . . . . . . . . . . . 17 ((𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛) → (¬ (𝑢𝑛) ≠ ∅ → (𝑢𝑋) ⊆ 𝑥))
69 toponss 22933 . . . . . . . . . . . . . . . . . . . . 21 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑢𝐾) → 𝑢𝑋)
7038, 69sylan 580 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → 𝑢𝑋)
71 dfss2 3969 . . . . . . . . . . . . . . . . . . . 20 (𝑢𝑋 ↔ (𝑢𝑋) = 𝑢)
7270, 71sylib 218 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → (𝑢𝑋) = 𝑢)
7372sseq1d 4015 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → ((𝑢𝑋) ⊆ 𝑥𝑢𝑥))
7473biimpd 229 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → ((𝑢𝑋) ⊆ 𝑥𝑢𝑥))
7568, 74syl9r 78 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → ((𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛) → (¬ (𝑢𝑛) ≠ ∅ → 𝑢𝑥)))
7663, 75syl5 34 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} → (¬ (𝑢𝑛) ≠ ∅ → 𝑢𝑥)))
7776rexlimdv 3153 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → (∃𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} ¬ (𝑢𝑛) ≠ ∅ → 𝑢𝑥))
7859, 77biimtrrid 243 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → (¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅ → 𝑢𝑥))
7978anim2d 612 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → ((𝑦𝑢 ∧ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅) → (𝑦𝑢𝑢𝑥)))
8079reximdva 3168 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → (∃𝑢𝐾 (𝑦𝑢 ∧ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅) → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥)))
8158, 80biimtrrid 243 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → (¬ ∀𝑢𝐾 (𝑦𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅) → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥)))
8257, 81mpd 15 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥))
8382anassrs 467 . . . . . . . 8 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) ∧ (𝑥𝑋𝑦𝑥)) → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥))
8483exp32 420 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → (𝑥𝑋 → (𝑦𝑥 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥))))
8529, 84pm2.61dne 3028 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → (𝑦𝑥 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥)))
8685ralrimiv 3145 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → ∀𝑦𝑥𝑢𝐾 (𝑦𝑢𝑢𝑥))
87 topontop 22919 . . . . . 6 (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
88 eltop2 22982 . . . . . 6 (𝐾 ∈ Top → (𝑥𝐾 ↔ ∀𝑦𝑥𝑢𝐾 (𝑦𝑢𝑢𝑥)))
8914, 87, 883syl 18 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → (𝑥𝐾 ↔ ∀𝑦𝑥𝑢𝐾 (𝑦𝑢𝑢𝑥)))
9086, 89mpbird 257 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → 𝑥𝐾)
9190ex 412 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) → (𝑥𝐽𝑥𝐾))
9291ssrdv 3989 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) → 𝐽𝐾)
9313, 92impbida 801 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (𝐽𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  cdif 3948  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  cfv 6561  (class class class)co 7431  Topctop 22899  TopOnctopon 22916  Filcfil 23853   fClus cfcls 23944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-topgen 17488  df-fbas 21361  df-top 22900  df-topon 22917  df-cld 23027  df-ntr 23028  df-cls 23029  df-fil 23854  df-fcls 23949
This theorem is referenced by: (None)
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