Theorem kqcldsat 23762

Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 23746). (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 23754	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3		elpreima 7091	. . . . . 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 4360	. . . . . . . 8 ⊢ ¬ (𝐹‘𝑧) ∈ ∅
7		elin 3992	. . . . . . . . 9 ⊢ ((𝐹‘𝑧) ∈ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) ↔ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) ∧ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
8		incom 4230	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈))
9		eqid 2740	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	cldss 23058	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ (Clsd‘𝐽) → 𝑈 ⊆ ∪ 𝐽)
11	10	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ ∪ 𝐽)
12		fndm 6682	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
13	2, 12	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝑋)
14		toponuni 22941	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
15	13, 14	eqtrd 2780	. . . . . . . . . . . . . . . . . . 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 4050	. . . . . . . . . . . . . . . . 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 4049	. . . . . . . . . . . . . . . 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 4288	. . . . . . . . . . . . . . 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 6089	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈))) = (𝐹 “ 𝑈))
24	23	ineq2d 4241	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ (𝑋 ∖ (𝑋 ∖ 𝑈)))) = ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈)))
25		simpll 766	. . . . . . . . . . . . 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 4148	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑈) = (∪ 𝐽 ∖ 𝑈))
28	9	cldopn 23060	. . . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . 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 23761	. . . . . . . . . . . . 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 2782	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ (𝑋 ∖ 𝑈)) ∩ (𝐹 “ 𝑈)) = ∅)
35	8, 34	eqtrid 2792	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅)
36	35	eleq2d 2830	. . . . . . . . 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 3986	. . . . . . . . . 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 23759	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ 𝑈) ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ (𝑋 ∖ 𝑈) ↔ (𝐹‘𝑧) ∈ (𝐹 “ (𝑋 ∖ 𝑈))))
46	25, 31, 44, 45	syl3anc 1371	. . . . . . . 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 4014	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) ⊆ 𝑈)
53		sseqin2 4244	. . . 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 6184	. . 3 ⊢ (dom 𝐹 ∩ 𝑈) ⊆ (◡𝐹 “ (𝐹 “ 𝑈))
56	54, 55	eqsstrrdi 4064	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ (◡𝐹 “ (𝐹 “ 𝑈)))
57	52, 56	eqssd 4026	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {crab 3443   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ∪ cuni 4931   ↦ cmpt 5249  ^◡ccnv 5699  dom cdm 5700   “ cima 5703   Fn wfn 6568  ‘cfv 6573  TopOnctopon 22937  Clsdccld 23045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-top 22921  df-topon 22938  df-cld 23048

This theorem is referenced by:  kqcld  23764