Theorem kqcldsat 23698

Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 23682). (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 23690	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3		elpreima 7011	. . . . . 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 4279	. . . . . . . 8 ⊢ ¬ (𝐹‘𝑧) ∈ ∅
7		elin 3906	. . . . . . . . 9 ⊢ ((𝐹‘𝑧) ∈ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
8		incom 4150	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈))
9		eqid 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	cldss 22994	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ (Clsd‘𝐽) → 𝑈 ⊆ ∪ 𝐽)
11	10	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ ∪ 𝐽)
12		fndm 6602	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
13	2, 12	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝑋)
14		toponuni 22879	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
15	13, 14	eqtrd 2772	. . . . . . . . . . . . . . . . . . 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 3960	. . . . . . . . . . . . . . . . 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 3959	. . . . . . . . . . . . . . . 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 4210	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑈)) = 𝑈)
22	20, 21	sylib 218	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝑋 ∖ (𝑋 ∖ 𝑈)) = 𝑈)
23	22	imaeq2d 6026	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈))) = (𝐹 “ 𝑈))
24	23	ineq2d 4161	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈)))) = ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈)))
25		simpll 767	. . . . . . . . . . . . 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 4066	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑈) = (∪ 𝐽 ∖ 𝑈))
28	9	cldopn 22996	. . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . 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 23697	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ 𝑈) ∈ 𝐽) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈)))) = ∅)
33	25, 31, 32	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈)))) = ∅)
34	24, 33	eqtr3d 2774	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈)) = ∅)
35	8, 34	eqtrid 2784	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅)
36	35	eleq2d 2823	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ (𝐹‘𝑧) ∈ ∅))
37	7, 36	bitr3id 285	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ (𝐹‘𝑧) ∈ ∅))
38	6, 37	mtbiri 327	. . . . . . 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 234	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) → ¬ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
41		eldif 3900	. . . . . . . . . 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 23695	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ 𝑈) ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ (𝑋 ∖ 𝑈) ↔ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
46	25, 31, 44, 45	syl3anc 1374	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ (𝑋 ∖ 𝑈) ↔ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
47	43, 46	bitrd 279	. . . . . . 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 239	. . . . 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 240	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) → 𝑧 ∈ 𝑈))
52	51	ssrdv 3928	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) ⊆ 𝑈)
53		sseqin2 4164	. . . 4 ⊢ (𝑈 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑈) = 𝑈)
54	17, 53	sylib 218	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (dom 𝐹 ∩ 𝑈) = 𝑈)
55		dminss 6118	. . 3 ⊢ (dom 𝐹 ∩ 𝑈) ⊆ (◡𝐹 “ (𝐹 “ 𝑈))
56	54, 55	eqsstrrdi 3968	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ (◡𝐹 “ (𝐹 “ 𝑈)))
57	52, 56	eqssd 3940	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3390   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  ∪ cuni 4851   ↦ cmpt 5167  ^◡ccnv 5630  dom cdm 5631   “ cima 5634   Fn wfn 6494  ‘cfv 6499  TopOnctopon 22875  Clsdccld 22981

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-top 22859  df-topon 22876  df-cld 22984

This theorem is referenced by:  kqcld  23700