Theorem kqtop 23683

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 22856	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
2		eqid 2735	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
3	2	kqtopon 23665	. . . 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 22851	. . 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 22842	. . . 4 ⊢ ((KQ‘𝐽) ∈ Top → ∅ ∈ (KQ‘𝐽))
8		elfvdm 6913	. . . 4 ⊢ (∅ ∈ (KQ‘𝐽) → 𝐽 ∈ dom KQ)
9	7, 8	syl 17	. . 3 ⊢ ((KQ‘𝐽) ∈ Top → 𝐽 ∈ dom KQ)
10		ovex 7438	. . . 4 ⊢ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})) ∈ V
11		df-kq 23632	. . . 4 ⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))
12	10, 11	dmmpti 6682	. . 3 ⊢ dom KQ = Top
13	9, 12	eleqtrdi 2844	. 2 ⊢ ((KQ‘𝐽) ∈ Top → 𝐽 ∈ Top)
14	6, 13	impbii 209	1 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)

Syntax hints:   ↔ wb 206   ∈ wcel 2108  {crab 3415  ∅c0 4308  ∪ cuni 4883   ↦ cmpt 5201  dom cdm 5654  ran crn 5655  ‘cfv 6531  (class class class)co 7405   qTop cqtop 17517  Topctop 22831  TopOnctopon 22848  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-kq 23632

This theorem is referenced by:  kqt0  23684  kqreg  23689  kqnrm  23690