Theorem kqcldsat 23901

Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 23885). (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 23893	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3		elpreima 7053	. . . . . 6 ⊢ (𝐹 Fn 𝑋 → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈))))
4	2, 3	syl 18	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈))))
5	4	adantr 485	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈))))
6		noel 4290	. . . . . . . 8 ⊢ ¬ (𝐹‘𝑧) ∈ ∅
7		elin 3920	. . . . . . . . 9 ⊢ ((𝐹‘𝑧) ∈ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
8		incom 4161	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈))
9		eqid 2762	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	cldss 23197	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ (Clsd‘𝐽) → 𝑈 ⊆ ∪ 𝐽)
11	10	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ ∪ 𝐽)
12		fndm 6638	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
13	2, 12	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝑋)
14		toponuni 23082	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
15	13, 14	eqtrd 2797	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = ∪ 𝐽)
16	15	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = ∪ 𝐽)
17	11, 16	sseqtrrd 3973	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ dom 𝐹)
18	13	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = 𝑋)
19	17, 18	sseqtrd 3972	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ 𝑋)
20	19	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → 𝑈 ⊆ 𝑋)
21		dfss4 4221	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑈)) = 𝑈)
22	20, 21	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝑋 ∖ (𝑋 ∖ 𝑈)) = 𝑈)
23	22	imaeq2d 6061	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈))) = (𝐹 “ 𝑈))
24	23	ineq2d 4172	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈)))) = ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈)))
25		simpll 778	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
26	14	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑋 = ∪ 𝐽)
27	26	difeq1d 4079	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑈) = (∪ 𝐽 ∖ 𝑈))
28	9	cldopn 23199	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝑈) ∈ 𝐽)
29	28	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑈) ∈ 𝐽)
30	27, 29	eqeltrd 2862	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑈) ∈ 𝐽)
31	30	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝑋 ∖ 𝑈) ∈ 𝐽)
32	1	kqdisj 23900	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ 𝑈) ∈ 𝐽) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈)))) = ∅)
33	25, 31, 32	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈)))) = ∅)
34	24, 33	eqtr3d 2799	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈)) = ∅)
35	8, 34	eqtrid 2809	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅)
36	35	eleq2d 2848	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ (𝐹‘𝑧) ∈ ∅))
37	7, 36	bitr3id 288	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ (𝐹‘𝑧) ∈ ∅))
38	6, 37	mtbiri 330	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ¬ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
39		imnan 404	. . . . . . 7 ⊢ (((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) → ¬ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ ¬ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
40	38, 39	sylibr 237	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) → ¬ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
41		eldif 3914	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 ∖ 𝑈) ↔ (𝑧 ∈ 𝑋 ∧ ¬ 𝑧 ∈ 𝑈))
42	41	baibr 545	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑋 → (¬ 𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (𝑋 ∖ 𝑈)))
43	42	adantl 486	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (¬ 𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (𝑋 ∖ 𝑈)))
44		simpr 489	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
45	1	kqfvima 23898	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ 𝑈) ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ (𝑋 ∖ 𝑈) ↔ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
46	25, 31, 44, 45	syl3anc 1397	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ (𝑋 ∖ 𝑈) ↔ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
47	43, 46	bitrd 282	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (¬ 𝑧 ∈ 𝑈 ↔ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
48	47	con1bid 358	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (¬ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ↔ 𝑧 ∈ 𝑈))
49	40, 48	sylibd 242	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) → 𝑧 ∈ 𝑈))
50	49	expimpd 458	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ((𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈)) → 𝑧 ∈ 𝑈))
51	5, 50	sylbid 243	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) → 𝑧 ∈ 𝑈))
52	51	ssrdv 3942	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) ⊆ 𝑈)
53		sseqin2 4175	. . . 4 ⊢ (𝑈 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑈) = 𝑈)
54	17, 53	sylib 221	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (dom 𝐹 ∩ 𝑈) = 𝑈)
55		dminss 6149	. . 3 ⊢ (dom 𝐹 ∩ 𝑈) ⊆ (◡𝐹 “ (𝐹 “ 𝑈))
56	54, 55	eqsstrrdi 3981	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ (◡𝐹 “ (𝐹 “ 𝑈)))
57	52, 56	eqssd 3953	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {crab 3415   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  ∪ cuni 4871   ↦ cmpt 5191  ^◡ccnv 5659  dom cdm 5660   “ cima 5663   Fn wfn 6531  ‘cfv 6536  TopOnctopon 23078  Clsdccld 23184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-top 23062  df-topon 23079  df-cld 23187

This theorem is used by:  kqcld  23903