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Theorem fclscf 22633
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 766 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐽 ∈ (TopOn‘𝑋))
2 simplr 768 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐾 ∈ (TopOn‘𝑋))
3 fclstopon 22620 . . . . . . . . 9 (𝑥 ∈ (𝐾 fClus 𝑓) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝑓 ∈ (Fil‘𝑋)))
43ad2antll 728 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝑓 ∈ (Fil‘𝑋)))
52, 4mpbid 235 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑓 ∈ (Fil‘𝑋))
6 simprl 770 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐽𝐾)
7 fclsss1 22630 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
81, 5, 6, 7syl3anc 1368 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
9 simprr 772 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑥 ∈ (𝐾 fClus 𝑓))
108, 9sseldd 3954 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽𝐾𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑥 ∈ (𝐽 fClus 𝑓))
1110expr 460 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) → (𝑥 ∈ (𝐾 fClus 𝑓) → 𝑥 ∈ (𝐽 fClus 𝑓)))
1211ssrdv 3959 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
1312ralrimivw 3178 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) → ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
14 simpllr 775 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → 𝐾 ∈ (TopOn‘𝑋))
15 toponmax 21534 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑋) → 𝑋𝐾)
16 ssid 3975 . . . . . . . . . . 11 𝑋𝑋
17 eleq2 2904 . . . . . . . . . . . . 13 (𝑢 = 𝑋 → (𝑦𝑢𝑦𝑋))
18 sseq1 3978 . . . . . . . . . . . . 13 (𝑢 = 𝑋 → (𝑢𝑋𝑋𝑋))
1917, 18anbi12d 633 . . . . . . . . . . . 12 (𝑢 = 𝑋 → ((𝑦𝑢𝑢𝑋) ↔ (𝑦𝑋𝑋𝑋)))
2019rspcev 3609 . . . . . . . . . . 11 ((𝑋𝐾 ∧ (𝑦𝑋𝑋𝑋)) → ∃𝑢𝐾 (𝑦𝑢𝑢𝑋))
2116, 20mpanr2 703 . . . . . . . . . 10 ((𝑋𝐾𝑦𝑋) → ∃𝑢𝐾 (𝑦𝑢𝑢𝑋))
2221ex 416 . . . . . . . . 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 2904 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑦𝑥𝑦𝑋))
25 sseq2 3979 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑢𝑥𝑢𝑋))
2625anbi2d 631 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝑦𝑢𝑢𝑥) ↔ (𝑦𝑢𝑢𝑋)))
2726rexbidv 3289 . . . . . . . . 9 (𝑥 = 𝑋 → (∃𝑢𝐾 (𝑦𝑢𝑢𝑥) ↔ ∃𝑢𝐾 (𝑦𝑢𝑢𝑋)))
2824, 27imbi12d 348 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑦𝑥 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥)) ↔ (𝑦𝑋 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑋))))
2923, 28syl5ibrcom 250 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → (𝑥 = 𝑋 → (𝑦𝑥 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥))))
30 simplll 774 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝐽 ∈ (TopOn‘𝑋))
31 simprl 770 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑥𝐽)
32 simprrr 781 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑦𝑥)
33 supnfcls 22628 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽𝑦𝑥) → ¬ 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}))
3430, 31, 32, 33syl3anc 1368 . . . . . . . . . . 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 21535 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
3630, 31, 35syl2anc 587 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑥𝑋)
3736, 32sseldd 3954 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑦𝑋)
38 simpllr 775 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝐾 ∈ (TopOn‘𝑋))
39 toponmax 21534 . . . . . . . . . . . . . . . 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 4095 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → (𝑋𝑥) ⊆ 𝑋)
42 simprrl 780 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → 𝑥𝑋)
43 pssdifn0 4308 . . . . . . . . . . . . . . . 16 ((𝑥𝑋𝑥𝑋) → (𝑋𝑥) ≠ ∅)
4436, 42, 43syl2anc 587 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) → (𝑋𝑥) ≠ ∅)
45 supfil 22503 . . . . . . . . . . . . . . 15 ((𝑋𝐽 ∧ (𝑋𝑥) ⊆ 𝑋 ∧ (𝑋𝑥) ≠ ∅) → {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} ∈ (Fil‘𝑋))
4640, 41, 44, 45syl3anc 1368 . . . . . . . . . . . . . 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 22622 . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . 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 706 . . . . . . . . . . . 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 7157 . . . . . . . . . . . . . . 15 (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} → (𝐾 fClus 𝑓) = (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}))
51 oveq2 7157 . . . . . . . . . . . . . . 15 (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} → (𝐽 fClus 𝑓) = (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦}))
5250, 51sseq12d 3986 . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . 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 3611 . . . . . . . . . . . . 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 3952 . . . . . . . . . . . 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 263 . . . . . . . . . . 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 201 . . . . . . . . . 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 3257 . . . . . . . . . . 11 (∃𝑢𝐾 (𝑦𝑢 ∧ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅) ↔ ¬ ∀𝑢𝐾 (𝑦𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅))
59 rexnal 3232 . . . . . . . . . . . . . 14 (∃𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} ¬ (𝑢𝑛) ≠ ∅ ↔ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} (𝑢𝑛) ≠ ∅)
60 sseq2 3979 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑛 → ((𝑋𝑥) ⊆ 𝑦 ↔ (𝑋𝑥) ⊆ 𝑛))
6160elrab 3666 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} ↔ (𝑛 ∈ 𝒫 𝑋 ∧ (𝑋𝑥) ⊆ 𝑛))
62 sslin 4196 . . . . . . . . . . . . . . . . 17 ((𝑋𝑥) ⊆ 𝑛 → (𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛))
6361, 62simplbiim 508 . . . . . . . . . . . . . . . 16 (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ⊆ 𝑦} → (𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛))
64 ssn0 4337 . . . . . . . . . . . . . . . . . . . 20 (((𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛) ∧ (𝑢 ∩ (𝑋𝑥)) ≠ ∅) → (𝑢𝑛) ≠ ∅)
6564ex 416 . . . . . . . . . . . . . . . . . . 19 ((𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛) → ((𝑢 ∩ (𝑋𝑥)) ≠ ∅ → (𝑢𝑛) ≠ ∅))
6665necon1bd 3032 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛) → (¬ (𝑢𝑛) ≠ ∅ → (𝑢 ∩ (𝑋𝑥)) = ∅))
67 inssdif0 4312 . . . . . . . . . . . . . . . . . 18 ((𝑢𝑋) ⊆ 𝑥 ↔ (𝑢 ∩ (𝑋𝑥)) = ∅)
6866, 67syl6ibr 255 . . . . . . . . . . . . . . . . 17 ((𝑢 ∩ (𝑋𝑥)) ⊆ (𝑢𝑛) → (¬ (𝑢𝑛) ≠ ∅ → (𝑢𝑋) ⊆ 𝑥))
69 toponss 21535 . . . . . . . . . . . . . . . . . . . . 21 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑢𝐾) → 𝑢𝑋)
7038, 69sylan 583 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → 𝑢𝑋)
71 df-ss 3936 . . . . . . . . . . . . . . . . . . . 20 (𝑢𝑋 ↔ (𝑢𝑋) = 𝑢)
7270, 71sylib 221 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → (𝑢𝑋) = 𝑢)
7372sseq1d 3984 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥𝐽 ∧ (𝑥𝑋𝑦𝑥))) ∧ 𝑢𝐾) → ((𝑢𝑋) ⊆ 𝑥𝑢𝑥))
7473biimpd 232 . . . . . . . . . . . . . . . . 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 3275 . . . . . . . . . . . . . 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, 77syl5bir 246 . . . . . . . . . . . . 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 614 . . . . . . . . . . . 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 3266 . . . . . . . . . . 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, 80syl5bir 246 . . . . . . . . . 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 471 . . . . . . . 8 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) ∧ (𝑥𝑋𝑦𝑥)) → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥))
8483exp32 424 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → (𝑥𝑋 → (𝑦𝑥 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥))))
8529, 84pm2.61dne 3100 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → (𝑦𝑥 → ∃𝑢𝐾 (𝑦𝑢𝑢𝑥)))
8685ralrimiv 3176 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → ∀𝑦𝑥𝑢𝐾 (𝑦𝑢𝑢𝑥))
87 topontop 21521 . . . . . 6 (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
88 eltop2 21583 . . . . . 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 260 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥𝐽) → 𝑥𝐾)
9190ex 416 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) → (𝑥𝐽𝑥𝐾))
9291ssrdv 3959 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) → 𝐽𝐾)
9313, 92impbida 800 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 209  wa 399   = wceq 1538  wcel 2115  wne 3014  wral 3133  wrex 3134  {crab 3137  cdif 3916  cin 3918  wss 3919  c0 4276  𝒫 cpw 4522  cfv 6343  (class class class)co 7149  Topctop 21501  TopOnctopon 21518  Filcfil 22453   fClus cfcls 22544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-topgen 16717  df-fbas 20542  df-top 21502  df-topon 21519  df-cld 21627  df-ntr 21628  df-cls 21629  df-fil 22454  df-fcls 22549
This theorem is referenced by: (None)
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