Theorem kqcldsat 22792

Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 22776). (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
kqcldsat	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈)

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦

Allowed substitution hints:   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem kqcldsat

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqffn 22784	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3		elpreima 6917	. . . . . 6 ⊢ (𝐹 Fn 𝑋 → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈))))
4	2, 3	syl 17	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈))))
5	4	adantr 480	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈))))
6		noel 4261	. . . . . . . 8 ⊢ ¬ (𝐹‘𝑧) ∈ ∅
7		elin 3899	. . . . . . . . 9 ⊢ ((𝐹‘𝑧) ∈ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
8		incom 4131	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈))
9		eqid 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	cldss 22088	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ (Clsd‘𝐽) → 𝑈 ⊆ ∪ 𝐽)
11	10	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ ∪ 𝐽)
12		fndm 6520	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
13	2, 12	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝑋)
14		toponuni 21971	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
15	13, 14	eqtrd 2778	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = ∪ 𝐽)
16	15	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = ∪ 𝐽)
17	11, 16	sseqtrrd 3958	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ dom 𝐹)
18	13	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = 𝑋)
19	17, 18	sseqtrd 3957	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ 𝑋)
20	19	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → 𝑈 ⊆ 𝑋)
21		dfss4 4189	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑈)) = 𝑈)
22	20, 21	sylib 217	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝑋 ∖ (𝑋 ∖ 𝑈)) = 𝑈)
23	22	imaeq2d 5958	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈))) = (𝐹 “ 𝑈))
24	23	ineq2d 4143	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈)))) = ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈)))
25		simpll 763	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
26	14	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑋 = ∪ 𝐽)
27	26	difeq1d 4052	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑈) = (∪ 𝐽 ∖ 𝑈))
28	9	cldopn 22090	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝑈) ∈ 𝐽)
29	28	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑈) ∈ 𝐽)
30	27, 29	eqeltrd 2839	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑈) ∈ 𝐽)
31	30	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝑋 ∖ 𝑈) ∈ 𝐽)
32	1	kqdisj 22791	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ 𝑈) ∈ 𝐽) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈)))) = ∅)
33	25, 31, 32	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈)))) = ∅)
34	24, 33	eqtr3d 2780	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈)) = ∅)
35	8, 34	eqtrid 2790	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅)
36	35	eleq2d 2824	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ (𝐹‘𝑧) ∈ ∅))
37	7, 36	bitr3id 284	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ (𝐹‘𝑧) ∈ ∅))
38	6, 37	mtbiri 326	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ¬ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
39		imnan 399	. . . . . . 7 ⊢ (((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) → ¬ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ ¬ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
40	38, 39	sylibr 233	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) → ¬ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
41		eldif 3893	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 ∖ 𝑈) ↔ (𝑧 ∈ 𝑋 ∧ ¬ 𝑧 ∈ 𝑈))
42	41	baibr 536	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑋 → (¬ 𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (𝑋 ∖ 𝑈)))
43	42	adantl 481	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (¬ 𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (𝑋 ∖ 𝑈)))
44		simpr 484	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
45	1	kqfvima 22789	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ 𝑈) ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ (𝑋 ∖ 𝑈) ↔ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
46	25, 31, 44, 45	syl3anc 1369	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ (𝑋 ∖ 𝑈) ↔ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
47	43, 46	bitrd 278	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (¬ 𝑧 ∈ 𝑈 ↔ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
48	47	con1bid 355	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (¬ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ↔ 𝑧 ∈ 𝑈))
49	40, 48	sylibd 238	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) → 𝑧 ∈ 𝑈))
50	49	expimpd 453	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ((𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈)) → 𝑧 ∈ 𝑈))
51	5, 50	sylbid 239	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) → 𝑧 ∈ 𝑈))
52	51	ssrdv 3923	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) ⊆ 𝑈)
53		sseqin2 4146	. . . 4 ⊢ (𝑈 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑈) = 𝑈)
54	17, 53	sylib 217	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (dom 𝐹 ∩ 𝑈) = 𝑈)
55		dminss 6045	. . 3 ⊢ (dom 𝐹 ∩ 𝑈) ⊆ (◡𝐹 “ (𝐹 “ 𝑈))
56	54, 55	eqsstrrdi 3972	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ (◡𝐹 “ (𝐹 “ 𝑈)))
57	52, 56	eqssd 3934	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3067   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836   ↦ cmpt 5153  ^◡ccnv 5579  dom cdm 5580   “ cima 5583   Fn wfn 6413  ‘cfv 6418  TopOnctopon 21967  Clsdccld 22075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-top 21951  df-topon 21968  df-cld 22078

This theorem is referenced by:  kqcld  22794