Theorem kqcldsat 22341

Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 22325). (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 22333	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3		elpreima 6828	. . . . . 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 483	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈))))
6		noel 4296	. . . . . . . 8 ⊢ ¬ (𝐹‘𝑧) ∈ ∅
7		elin 4169	. . . . . . . . 9 ⊢ ((𝐹‘𝑧) ∈ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
8		incom 4178	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈))
9		eqid 2821	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	cldss 21637	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ (Clsd‘𝐽) → 𝑈 ⊆ ∪ 𝐽)
11	10	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ ∪ 𝐽)
12		fndm 6455	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
13	2, 12	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝑋)
14		toponuni 21522	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
15	13, 14	eqtrd 2856	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = ∪ 𝐽)
16	15	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = ∪ 𝐽)
17	11, 16	sseqtrrd 4008	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ dom 𝐹)
18	13	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom 𝐹 = 𝑋)
19	17, 18	sseqtrd 4007	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ 𝑋)
20	19	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → 𝑈 ⊆ 𝑋)
21		dfss4 4235	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑈)) = 𝑈)
22	20, 21	sylib 220	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝑋 ∖ (𝑋 ∖ 𝑈)) = 𝑈)
23	22	imaeq2d 5929	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈))) = (𝐹 “ 𝑈))
24	23	ineq2d 4189	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈)))) = ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈)))
25		simpll 765	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
26	14	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑋 = ∪ 𝐽)
27	26	difeq1d 4098	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑈) = (∪ 𝐽 ∖ 𝑈))
28	9	cldopn 21639	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝑈) ∈ 𝐽)
29	28	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑈) ∈ 𝐽)
30	27, 29	eqeltrd 2913	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑈) ∈ 𝐽)
31	30	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝑋 ∖ 𝑈) ∈ 𝐽)
32	1	kqdisj 22340	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ 𝑈) ∈ 𝐽) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈)))) = ∅)
33	25, 31, 32	syl2anc 586	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈)))) = ∅)
34	24, 33	eqtr3d 2858	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈)) = ∅)
35	8, 34	syl5eq 2868	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅)
36	35	eleq2d 2898	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ (𝐹‘𝑧) ∈ ∅))
37	7, 36	syl5bbr 287	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ (𝐹‘𝑧) ∈ ∅))
38	6, 37	mtbiri 329	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ¬ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
39		imnan 402	. . . . . . 7 ⊢ (((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) → ¬ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ ¬ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
40	38, 39	sylibr 236	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) → ¬ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
41		eldif 3946	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 ∖ 𝑈) ↔ (𝑧 ∈ 𝑋 ∧ ¬ 𝑧 ∈ 𝑈))
42	41	baibr 539	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑋 → (¬ 𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (𝑋 ∖ 𝑈)))
43	42	adantl 484	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (¬ 𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (𝑋 ∖ 𝑈)))
44		simpr 487	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
45	1	kqfvima 22338	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ 𝑈) ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ (𝑋 ∖ 𝑈) ↔ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
46	25, 31, 44, 45	syl3anc 1367	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ (𝑋 ∖ 𝑈) ↔ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
47	43, 46	bitrd 281	. . . . . . 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 241	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) → 𝑧 ∈ 𝑈))
50	49	expimpd 456	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ((𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈)) → 𝑧 ∈ 𝑈))
51	5, 50	sylbid 242	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) → 𝑧 ∈ 𝑈))
52	51	ssrdv 3973	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) ⊆ 𝑈)
53		sseqin2 4192	. . . 4 ⊢ (𝑈 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑈) = 𝑈)
54	17, 53	sylib 220	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (dom 𝐹 ∩ 𝑈) = 𝑈)
55		dminss 6010	. . 3 ⊢ (dom 𝐹 ∩ 𝑈) ⊆ (◡𝐹 “ (𝐹 “ 𝑈))
56	54, 55	eqsstrrdi 4022	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ (◡𝐹 “ (𝐹 “ 𝑈)))
57	52, 56	eqssd 3984	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {crab 3142   ∖ cdif 3933   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  ∪ cuni 4838   ↦ cmpt 5146  ^◡ccnv 5554  dom cdm 5555   “ cima 5558   Fn wfn 6350  ‘cfv 6355  TopOnctopon 21518  Clsdccld 21624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-top 21502  df-topon 21519  df-cld 21627

This theorem is referenced by:  kqcld  22343