Theorem kqtop 23753

Description: The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)

Assertion

Ref	Expression
kqtop	⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)

Proof of Theorem kqtop

Dummy variables 𝑥 𝑦 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		toptopon2 22924	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
2		eqid 2737	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
3	2	kqtopon 23735	. . . 4 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})))
4	1, 3	sylbi 217	. . 3 ⊢ (𝐽 ∈ Top → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})))
5		topontop 22919	. . 3 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})) → (KQ‘𝐽) ∈ Top)
6	4, 5	syl 17	. 2 ⊢ (𝐽 ∈ Top → (KQ‘𝐽) ∈ Top)
7		0opn 22910	. . . 4 ⊢ ((KQ‘𝐽) ∈ Top → ∅ ∈ (KQ‘𝐽))
8		elfvdm 6943	. . . 4 ⊢ (∅ ∈ (KQ‘𝐽) → 𝐽 ∈ dom KQ)
9	7, 8	syl 17	. . 3 ⊢ ((KQ‘𝐽) ∈ Top → 𝐽 ∈ dom KQ)
10		ovex 7464	. . . 4 ⊢ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})) ∈ V
11		df-kq 23702	. . . 4 ⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))
12	10, 11	dmmpti 6712	. . 3 ⊢ dom KQ = Top
13	9, 12	eleqtrdi 2851	. 2 ⊢ ((KQ‘𝐽) ∈ Top → 𝐽 ∈ Top)
14	6, 13	impbii 209	1 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)

Syntax hints:   ↔ wb 206   ∈ wcel 2108  {crab 3436  ∅c0 4333  ∪ cuni 4907   ↦ cmpt 5225  dom cdm 5685  ran crn 5686  ‘cfv 6561  (class class class)co 7431   qTop cqtop 17548  Topctop 22899  TopOnctopon 22916  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-kq 23702

This theorem is referenced by:  kqt0  23754  kqreg  23759  kqnrm  23760