Theorem fclscf 24150

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)) → (𝐽 ⊆ 𝐾 ↔ ∀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.

Step	Hyp	Ref	Expression
1		simpll 778	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐽 ∈ (TopOn‘𝑋))
2		simplr 780	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐾 ∈ (TopOn‘𝑋))
3		fclstopon 24137	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐾 fClus 𝑓) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝑓 ∈ (Fil‘𝑋)))
4	3	ad2antll 741	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝑓 ∈ (Fil‘𝑋)))
5	2, 4	mpbid 235	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑓 ∈ (Fil‘𝑋))
6		simprl 782	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐽 ⊆ 𝐾)
7		fclsss1 24147	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
8	1, 5, 6, 7	syl3anc 1396	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
9		simprr 784	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑥 ∈ (𝐾 fClus 𝑓))
10	8, 9	sseldd 3946	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑥 ∈ (𝐽 fClus 𝑓))
11	10	expr 461	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝐾 fClus 𝑓) → 𝑥 ∈ (𝐽 fClus 𝑓)))
12	11	ssrdv 3951	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
13	12	ralrimivw 3167	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) → ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
14		simpllr 787	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑋))
15		toponmax 23051	. . . . . . . . 9 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐾)
16		ssid 3967	. . . . . . . . . . 11 ⊢ 𝑋 ⊆ 𝑋
17		eleq2 2858	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑋 → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ 𝑋))
18		sseq1 3970	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑋 → (𝑢 ⊆ 𝑋 ↔ 𝑋 ⊆ 𝑋))
19	17, 18	anbi12d 643	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑋 → ((𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋) ↔ (𝑦 ∈ 𝑋 ∧ 𝑋 ⊆ 𝑋)))
20	19	rspcev 3590	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐾 ∧ (𝑦 ∈ 𝑋 ∧ 𝑋 ⊆ 𝑋)) → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋))
21	16, 20	mpanr2 716	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋))
22	21	ex 417	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐾 → (𝑦 ∈ 𝑋 → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋)))
23	14, 15, 22	3syl 19	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → (𝑦 ∈ 𝑋 → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋)))
24		eleq2 2858	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑋))
25		sseq2 3971	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑢 ⊆ 𝑥 ↔ 𝑢 ⊆ 𝑋))
26	25	anbi2d 641	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥) ↔ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋)))
27	26	rexbidv 3195	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥) ↔ ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋)))
28	24, 27	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑦 ∈ 𝑥 → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥)) ↔ (𝑦 ∈ 𝑋 → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋))))
29	23, 28	syl5ibrcom 250	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → (𝑥 = 𝑋 → (𝑦 ∈ 𝑥 → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥))))
30		simplll 786	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝐽 ∈ (TopOn‘𝑋))
31		simprl 782	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝑥 ∈ 𝐽)
32		simprrr 793	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝑦 ∈ 𝑥)
33		supnfcls 24145	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ¬ 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}))
34	30, 31, 32, 33	syl3anc 1396	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → ¬ 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}))
35		toponss 23052	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
36	30, 31, 35	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝑥 ⊆ 𝑋)
37	36, 32	sseldd 3946	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝑦 ∈ 𝑋)
38		simpllr 787	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝐾 ∈ (TopOn‘𝑋))
39		toponmax 23051	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
40	30, 39	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝑋 ∈ 𝐽)
41		difssd 4099	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (𝑋 ∖ 𝑥) ⊆ 𝑋)
42		simprrl 792	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝑥 ≠ 𝑋)
43		pssdifn0 4331	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑥 ≠ 𝑋) → (𝑋 ∖ 𝑥) ≠ ∅)
44	36, 42, 43	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (𝑋 ∖ 𝑥) ≠ ∅)
45		supfil 24020	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑋 ∖ 𝑥) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑥) ≠ ∅) → {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} ∈ (Fil‘𝑋))
46	40, 41, 44, 45	syl3anc 1396	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} ∈ (Fil‘𝑋))
47		fclsopn 24139	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} ∈ (Fil‘𝑋)) → (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}) ↔ (𝑦 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅))))
48	38, 46, 47	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}) ↔ (𝑦 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅))))
49	37, 48	mpbirand 719	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}) ↔ ∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅)))
50		oveq2 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} → (𝐾 fClus 𝑓) = (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}))
51		oveq2 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} → (𝐽 fClus 𝑓) = (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}))
52	50, 51	sseq12d 3978	. . . . . . . . . . . . . 14 ⊢ (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} → ((𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓) ↔ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}) ⊆ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦})))
53		simplr 780	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))
54	52, 53, 46	rspcdva 3591	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}) ⊆ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}))
55	54	sseld 3944	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}) → 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦})))
56	49, 55	sylbird 263	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅) → 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦})))
57	34, 56	mtod 201	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → ¬ ∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅))
58		rexanali 3125	. . . . . . . . . . 11 ⊢ (∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅) ↔ ¬ ∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅))
59		rexnal 3123	. . . . . . . . . . . . . 14 ⊢ (∃𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} ¬ (𝑢 ∩ 𝑛) ≠ ∅ ↔ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅)
60		sseq2 3971	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑛 → ((𝑋 ∖ 𝑥) ⊆ 𝑦 ↔ (𝑋 ∖ 𝑥) ⊆ 𝑛))
61	60	elrab 3659	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} ↔ (𝑛 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑥) ⊆ 𝑛))
62		sslin 4203	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∖ 𝑥) ⊆ 𝑛 → (𝑢 ∩ (𝑋 ∖ 𝑥)) ⊆ (𝑢 ∩ 𝑛))
63	61, 62	simplbiim 513	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} → (𝑢 ∩ (𝑋 ∖ 𝑥)) ⊆ (𝑢 ∩ 𝑛))
64		ssn0 4368	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑢 ∩ (𝑋 ∖ 𝑥)) ⊆ (𝑢 ∩ 𝑛) ∧ (𝑢 ∩ (𝑋 ∖ 𝑥)) ≠ ∅) → (𝑢 ∩ 𝑛) ≠ ∅)
65	64	ex 417	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∩ (𝑋 ∖ 𝑥)) ⊆ (𝑢 ∩ 𝑛) → ((𝑢 ∩ (𝑋 ∖ 𝑥)) ≠ ∅ → (𝑢 ∩ 𝑛) ≠ ∅))
66	65	necon1bd 2982	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∩ (𝑋 ∖ 𝑥)) ⊆ (𝑢 ∩ 𝑛) → (¬ (𝑢 ∩ 𝑛) ≠ ∅ → (𝑢 ∩ (𝑋 ∖ 𝑥)) = ∅))
67		inssdif0 4337	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∩ 𝑋) ⊆ 𝑥 ↔ (𝑢 ∩ (𝑋 ∖ 𝑥)) = ∅)
68	66, 67	imbitrrdi 255	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∩ (𝑋 ∖ 𝑥)) ⊆ (𝑢 ∩ 𝑛) → (¬ (𝑢 ∩ 𝑛) ≠ ∅ → (𝑢 ∩ 𝑋) ⊆ 𝑥))
69		toponss 23052	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑢 ∈ 𝐾) → 𝑢 ⊆ 𝑋)
70	38, 69	sylan 591	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → 𝑢 ⊆ 𝑋)
71		dfss2 3931	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 ⊆ 𝑋 ↔ (𝑢 ∩ 𝑋) = 𝑢)
72	70, 71	sylib 221	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → (𝑢 ∩ 𝑋) = 𝑢)
73	72	sseq1d 3976	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → ((𝑢 ∩ 𝑋) ⊆ 𝑥 ↔ 𝑢 ⊆ 𝑥))
74	73	biimpd 232	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → ((𝑢 ∩ 𝑋) ⊆ 𝑥 → 𝑢 ⊆ 𝑥))
75	68, 74	syl9r 79	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → ((𝑢 ∩ (𝑋 ∖ 𝑥)) ⊆ (𝑢 ∩ 𝑛) → (¬ (𝑢 ∩ 𝑛) ≠ ∅ → 𝑢 ⊆ 𝑥)))
76	63, 75	syl5 35	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} → (¬ (𝑢 ∩ 𝑛) ≠ ∅ → 𝑢 ⊆ 𝑥)))
77	76	rexlimdv 3170	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → (∃𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} ¬ (𝑢 ∩ 𝑛) ≠ ∅ → 𝑢 ⊆ 𝑥))
78	59, 77	biimtrrid 246	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → (¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅ → 𝑢 ⊆ 𝑥))
79	78	anim2d 623	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → ((𝑦 ∈ 𝑢 ∧ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅) → (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥)))
80	79	reximdva 3184	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅) → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥)))
81	58, 80	biimtrrid 246	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (¬ ∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅) → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥)))
82	57, 81	mpd 16	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥))
83	82	anassrs 472	. . . . . . . 8 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥)) → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥))
84	83	exp32 425	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → (𝑥 ≠ 𝑋 → (𝑦 ∈ 𝑥 → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥))))
85	29, 84	pm2.61dne 3050	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → (𝑦 ∈ 𝑥 → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥)))
86	85	ralrimiv 3162	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥))
87		topontop 23038	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
88		eltop2 23100	. . . . . 6 ⊢ (𝐾 ∈ Top → (𝑥 ∈ 𝐾 ↔ ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥)))
89	14, 87, 88	3syl 19	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∈ 𝐾 ↔ ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥)))
90	86, 89	mpbird 260	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐾)
91	90	ex 417	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) → (𝑥 ∈ 𝐽 → 𝑥 ∈ 𝐾))
92	91	ssrdv 3951	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) → 𝐽 ⊆ 𝐾)
93	13, 92	impbida 812	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  {crab 3423   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  ‘cfv 6537  (class class class)co 7411  Topctop 23018  TopOnctopon 23035  Filcfil 23970   fClus cfcls 24061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-topgen 17495  df-fbas 21487  df-top 23019  df-topon 23036  df-cld 23144  df-ntr 23145  df-cls 23146  df-fil 23971  df-fcls 24066

This theorem is referenced by: (None)