Theorem kqcld 23743

Description: The topological indistinguishability map is a closed map. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

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

Assertion

Ref	Expression
kqcld	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑈) ∈ (Clsd‘(KQ‘𝐽)))

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

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

Proof of Theorem kqcld

Step	Hyp	Ref	Expression
1		imassrn 6089	. . . 4 ⊢ (𝐹 “ 𝑈) ⊆ ran 𝐹
2	1	a1i 11	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑈) ⊆ ran 𝐹)
3		kqval.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
4	3	kqcldsat 23741	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈)
5		simpr 484	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ∈ (Clsd‘𝐽))
6	4, 5	eqeltrd 2841	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) ∈ (Clsd‘𝐽))
7	3	kqffn 23733	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
8		dffn4 6826	. . . . . 6 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
9	7, 8	sylib 218	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋–onto→ran 𝐹)
10		qtopcld 23721	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→ran 𝐹) → ((𝐹 “ 𝑈) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝐹 “ 𝑈) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑈)) ∈ (Clsd‘𝐽))))
11	9, 10	mpdan 687	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝐹 “ 𝑈) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝐹 “ 𝑈) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑈)) ∈ (Clsd‘𝐽))))
12	11	adantr 480	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → ((𝐹 “ 𝑈) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝐹 “ 𝑈) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑈)) ∈ (Clsd‘𝐽))))
13	2, 6, 12	mpbir2and 713	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑈) ∈ (Clsd‘(𝐽 qTop 𝐹)))
14	3	kqval 23734	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
15	14	adantr 480	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
16	15	fveq2d 6910	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (Clsd‘(KQ‘𝐽)) = (Clsd‘(𝐽 qTop 𝐹)))
17	13, 16	eleqtrrd 2844	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑈) ∈ (Clsd‘(KQ‘𝐽)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {crab 3436   ⊆ wss 3951   ↦ cmpt 5225  ^◡ccnv 5684  ran crn 5686   “ cima 5688   Fn wfn 6556  –onto→wfo 6559  ‘cfv 6561  (class class class)co 7431   qTop cqtop 17548  TopOnctopon 22916  Clsdccld 23024  KQckq 23701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-qtop 17552  df-top 22900  df-topon 22917  df-cld 23027  df-kq 23702

This theorem is referenced by:  kqreglem1  23749  kqnrmlem1  23751  kqnrmlem2  23752