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Theorem fclscf 22030
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 807 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐽 ∈ (TopOn‘𝑋))
2 simplr 809 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐾 ∈ (TopOn‘𝑋))
3 fclstopon 22017 . . . . . . . . 9 (𝑥 ∈ (𝐾 fClus 𝑓) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝑓 ∈ (Fil‘𝑋)))
43ad2antll 767 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝑓 ∈ (Fil‘𝑋)))
52, 4mpbid 222 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑓 ∈ (Fil‘𝑋))
6 simprl 811 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐽𝐾)
7 fclsss1 22027 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
81, 5, 6, 7syl3anc 1477 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
9 simprr 813 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑥 ∈ (𝐾 fClus 𝑓))
108, 9sseldd 3745 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑥 ∈ (𝐽 fClus 𝑓))
1110expr 644 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) → (𝑥 ∈ (𝐾 fClus 𝑓) → 𝑥 ∈ (𝐽 fClus 𝑓)))
1211ssrdv 3750 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
1312ralrimivw 3105 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) → ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
14 simpllr 817 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → 𝐾 ∈ (TopOn‘𝑋))
15 toponmax 20932 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑋) → 𝑋𝐾)
16 ssid 3765 . . . . . . . . . . 11 𝑋𝑋
17 eleq2 2828 . . . . . . . . . . . . 13 (𝑢 = 𝑋 → (𝑦𝑢𝑦𝑋))
18 sseq1 3767 . . . . . . . . . . . . 13 (𝑢 = 𝑋 → (𝑢𝑋𝑋𝑋))
1917, 18anbi12d 749 . . . . . . . . . . . 12 (𝑢 = 𝑋 → ((𝑦𝑢𝑢𝑋) ↔ (𝑦𝑋𝑋𝑋)))
2019rspcev 3449 . . . . . . . . . . 11 ((𝑋𝐾 ∧ (𝑦𝑋𝑋𝑋)) → ∃𝑢𝐾 (𝑦𝑢𝑢𝑋))
2116, 20mpanr2 722 . . . . . . . . . 10 ((𝑋𝐾𝑦𝑋) → ∃𝑢𝐾 (𝑦𝑢𝑢𝑋))
2221ex 449 . . . . . . . . 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 2828 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑦𝑥𝑦𝑋))
25 sseq2 3768 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑢𝑥𝑢𝑋))
2625anbi2d 742 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝑦𝑢𝑢𝑥) ↔ (𝑦𝑢𝑢𝑋)))
2726rexbidv 3190 . . . . . . . . 9 (𝑥 = 𝑋 → (∃𝑢𝐾 (𝑦𝑢𝑢𝑥) ↔ ∃𝑢𝐾 (𝑦𝑢𝑢𝑋)))
2824, 27imbi12d 333 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑦𝑥 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥)) ↔ (𝑦𝑋 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑋))))
2923, 28syl5ibrcom 237 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → (𝑥 = 𝑋 → (𝑦𝑥 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥))))
30 simplll 815 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝐽 ∈ (TopOn‘𝑋))
31 simprl 811 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑥𝐽)
32 simprrr 824 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑦𝑥)
33 supnfcls 22025 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽𝑦𝑥) → ¬ 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}))
3430, 31, 32, 33syl3anc 1477 . . . . . . . . . . 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 simpllr 817 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝐾 ∈ (TopOn‘𝑋))
36 toponmax 20932 . . . . . . . . . . . . . . . 16 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
3730, 36syl 17 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑋𝐽)
38 difssd 3881 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → (𝑋𝑥) ⊆ 𝑋)
39 toponss 20933 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
4030, 31, 39syl2anc 696 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑥𝑋)
41 simprrl 823 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑥𝑋)
42 pssdifn0 4087 . . . . . . . . . . . . . . . 16 ((𝑥𝑋𝑥𝑋) → (𝑋𝑥) ≠ ∅)
4340, 41, 42syl2anc 696 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → (𝑋𝑥) ≠ ∅)
44 supfil 21900 . . . . . . . . . . . . . . 15 ((𝑋𝐽 ∧ (𝑋𝑥) ⊆ 𝑋 ∧ (𝑋𝑥) ≠ ∅) → {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} ∈ (Fil‘𝑋))
4537, 38, 43, 44syl3anc 1477 . . . . . . . . . . . . . 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))
46 fclsopn 22019 . . . . . . . . . . . . . 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) ≠ ∅))))
4735, 45, 46syl2anc 696 . . . . . . . . . . . . 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) ≠ ∅))))
4840, 32sseldd 3745 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑦𝑋)
4948biantrurd 530 . . . . . . . . . . . . 13 ((((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) ≠ ∅) ↔ (𝑦𝑋 ∧ ∀𝑢𝐾 (𝑦𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅))))
5047, 49bitr4d 271 . . . . . . . . . . . 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) ≠ ∅)))
51 simplr 809 . . . . . . . . . . . . . 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))
52 oveq2 6821 . . . . . . . . . . . . . . . 16 (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} → (𝐾 fClus 𝑓) = (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}))
53 oveq2 6821 . . . . . . . . . . . . . . . 16 (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} → (𝐽 fClus 𝑓) = (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}))
5452, 53sseq12d 3775 . . . . . . . . . . . . . . 15 (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})))
5554rspcv 3445 . . . . . . . . . . . . . 14 ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} ∈ (Fil‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓) → (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}) ⊆ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦})))
5645, 51, 55sylc 65 . . . . . . . . . . . . 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}))
5756sseld 3743 . . . . . . . . . . . 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})))
5850, 57sylbird 250 . . . . . . . . . . 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})))
5934, 58mtod 189 . . . . . . . . . 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) ≠ ∅))
60 rexanali 3136 . . . . . . . . . . 11 (∃𝑢𝐾 (𝑦𝑢 ∧ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅) ↔ ¬ ∀𝑢𝐾 (𝑦𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅))
61 rexnal 3133 . . . . . . . . . . . . . 14 (∃𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} ¬ (𝑢𝑛) ≠ ∅ ↔ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅)
62 sseq2 3768 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑛 → ((𝑋𝑥) ⊆ 𝑦 ↔ (𝑋𝑥) ⊆ 𝑛))
6362elrab 3504 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} ↔ (𝑛 ∈ 𝒫 𝑋 ∧ (𝑋𝑥) ⊆ 𝑛))
6463simprbi 483 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} → (𝑋𝑥) ⊆ 𝑛)
65 sslin 3982 . . . . . . . . . . . . . . . . 17 ((𝑋𝑥) ⊆ 𝑛 → (𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛))
6664, 65syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} → (𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛))
67 ssn0 4119 . . . . . . . . . . . . . . . . . . . 20 (((𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛) ∧ (𝑢 ∩ (𝑋𝑥)) ≠ ∅) → (𝑢𝑛) ≠ ∅)
6867ex 449 . . . . . . . . . . . . . . . . . . 19 ((𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛) → ((𝑢 ∩ (𝑋𝑥)) ≠ ∅ → (𝑢𝑛) ≠ ∅))
6968necon1bd 2950 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛) → (¬ (𝑢𝑛) ≠ ∅ → (𝑢 ∩ (𝑋𝑥)) = ∅))
70 inssdif0 4090 . . . . . . . . . . . . . . . . . 18 ((𝑢𝑋) ⊆ 𝑥 ↔ (𝑢 ∩ (𝑋𝑥)) = ∅)
7169, 70syl6ibr 242 . . . . . . . . . . . . . . . . 17 ((𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛) → (¬ (𝑢𝑛) ≠ ∅ → (𝑢𝑋) ⊆ 𝑥))
72 toponss 20933 . . . . . . . . . . . . . . . . . . . . 21 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑢𝐾) → 𝑢𝑋)
7335, 72sylan 489 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → 𝑢𝑋)
74 df-ss 3729 . . . . . . . . . . . . . . . . . . . 20 (𝑢𝑋 ↔ (𝑢𝑋) = 𝑢)
7573, 74sylib 208 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → (𝑢𝑋) = 𝑢)
7675sseq1d 3773 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → ((𝑢𝑋) ⊆ 𝑥𝑢𝑥))
7776biimpd 219 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → ((𝑢𝑋) ⊆ 𝑥𝑢𝑥))
7871, 77syl9r 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)))
7966, 78syl5 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)))
8079rexlimdv 3168 . . . . . . . . . . . . . 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))
8161, 80syl5bir 233 . . . . . . . . . . . . 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))
8281anim2d 590 . . . . . . . . . . . 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)))
8382reximdva 3155 . . . . . . . . . . 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)))
8460, 83syl5bir 233 . . . . . . . . . 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)))
8559, 84mpd 15 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥))
8685anassrs 683 . . . . . . . 8 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) ∧ (𝑥𝑋𝑦𝑥)) → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥))
8786exp32 632 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → (𝑥𝑋 → (𝑦𝑥 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥))))
8829, 87pm2.61dne 3018 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → (𝑦𝑥 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥)))
8988ralrimiv 3103 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → ∀𝑦𝑥𝑢𝐾 (𝑦𝑢𝑢𝑥))
90 topontop 20920 . . . . . 6 (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
91 eltop2 20981 . . . . . 6 (𝐾 ∈ Top → (𝑥𝐾 ↔ ∀𝑦𝑥𝑢𝐾 (𝑦𝑢𝑢𝑥)))
9214, 90, 913syl 18 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → (𝑥𝐾 ↔ ∀𝑦𝑥𝑢𝐾 (𝑦𝑢𝑢𝑥)))
9389, 92mpbird 247 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → 𝑥𝐾)
9493ex 449 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) → (𝑥𝐽𝑥𝐾))
9594ssrdv 3750 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) → 𝐽𝐾)
9613, 95impbida 913 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 196  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  {crab 3054  cdif 3712  cin 3714  wss 3715  c0 4058  𝒫 cpw 4302  cfv 6049  (class class class)co 6813  Topctop 20900  TopOnctopon 20917  Filcfil 21850   fClus cfcls 21941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-topgen 16306  df-fbas 19945  df-top 20901  df-topon 20918  df-cld 21025  df-ntr 21026  df-cls 21027  df-fil 21851  df-fcls 21946
This theorem is referenced by: (None)
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