Theorem kqcld 23871

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 6073	. . . 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 23869	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈)
5		simpr 489	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ∈ (Clsd‘𝐽))
6	4, 5	eqeltrd 2861	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) ∈ (Clsd‘𝐽))
7	3	kqffn 23861	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
8		dffn4 6798	. . . . . 6 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
9	7, 8	sylib 221	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋–onto→ran 𝐹)
10		qtopcld 23849	. . . . 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 699	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝐹 “ 𝑈) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝐹 “ 𝑈) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑈)) ∈ (Clsd‘𝐽))))
12	11	adantr 485	. . 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 725	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑈) ∈ (Clsd‘(𝐽 qTop 𝐹)))
14	3	kqval 23862	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
15	14	adantr 485	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
16	15	fveq2d 6885	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (Clsd‘(KQ‘𝐽)) = (Clsd‘(𝐽 qTop 𝐹)))
17	13, 16	eleqtrrd 2864	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑈) ∈ (Clsd‘(KQ‘𝐽)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  {crab 3414   ⊆ wss 3904   ↦ cmpt 5191  ^◡ccnv 5660  ran crn 5662   “ cima 5664   Fn wfn 6531  –onto→wfo 6534  ‘cfv 6536  (class class class)co 7410   qTop cqtop 17556  TopOnctopon 23046  Clsdccld 23152  KQckq 23829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  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-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-qtop 17560  df-top 23030  df-topon 23047  df-cld 23155  df-kq 23830

This theorem is referenced by:  kqreglem1  23877  kqnrmlem1  23879  kqnrmlem2  23880