Theorem kqtop 24064

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 23236	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
2		eqid 2761	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
3	2	kqtopon 24046	. . . 4 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})))
4	1, 3	sylbi 220	. . 3 ⊢ (𝐽 ∈ Top → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})))
5		topontop 23231	. . 3 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})) → (KQ‘𝐽) ∈ Top)
6	4, 5	syl 18	. 2 ⊢ (𝐽 ∈ Top → (KQ‘𝐽) ∈ Top)
7		0opn 23222	. . . 4 ⊢ ((KQ‘𝐽) ∈ Top → ∅ ∈ (KQ‘𝐽))
8		elfvdm 6919	. . . 4 ⊢ (∅ ∈ (KQ‘𝐽) → 𝐽 ∈ dom KQ)
9	7, 8	syl 18	. . 3 ⊢ ((KQ‘𝐽) ∈ Top → 𝐽 ∈ dom KQ)
10		ovex 7453	. . . 4 ⊢ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})) ∈ V
11		df-kq 24013	. . . 4 ⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))
12	10, 11	dmmpti 6683	. . 3 ⊢ dom KQ = Top
13	9, 12	eleqtrdi 2871	. 2 ⊢ ((KQ‘𝐽) ∈ Top → 𝐽 ∈ Top)
14	6, 13	impbii 212	1 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2145  {crab 3413  ∅c0 4279  ∪ cuni 4867   ↦ cmpt 5186  dom cdm 5651  ran crn 5652  ‘cfv 6538  (class class class)co 7420   qTop cqtop 17675  Topctop 23211  TopOnctopon 23228  KQckq 24012

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-qtop 17679  df-top 23212  df-topon 23229  df-kq 24013

This theorem is used by:  kqt0  24065  kqreg  24070  kqnrm  24071