Theorem kqcld 23673

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 6058	. . . 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 23671	. . . 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 2834	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) ∈ (Clsd‘𝐽))
7	3	kqffn 23663	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
8		dffn4 6796	. . . . . 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 23651	. . . . 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 23664	. . . 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 6880	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (Clsd‘(KQ‘𝐽)) = (Clsd‘(𝐽 qTop 𝐹)))
17	13, 16	eleqtrrd 2837	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 3415   ⊆ wss 3926   ↦ cmpt 5201  ^◡ccnv 5653  ran crn 5655   “ cima 5657   Fn wfn 6526  –onto→wfo 6529  ‘cfv 6531  (class class class)co 7405   qTop cqtop 17517  TopOnctopon 22848  Clsdccld 22954  KQckq 23631

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-qtop 17521  df-top 22832  df-topon 22849  df-cld 22957  df-kq 23632

This theorem is referenced by:  kqreglem1  23679  kqnrmlem1  23681  kqnrmlem2  23682