Theorem kqtopon 23669

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

Hypothesis

Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
kqtopon	⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem kqtopon

Step	Hyp	Ref	Expression
1		kqval.2	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqval 23668	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
3	1	kqffn 23667	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
4		dffn4 6750	. . . 4 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
5	3, 4	sylib 218	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋–onto→ran 𝐹)
6		qtoptopon 23646	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→ran 𝐹) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
7	5, 6	mpdan 687	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
8	2, 7	eqeltrd 2834	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {crab 3397   ↦ cmpt 5177  ran crn 5623   Fn wfn 6485  –onto→wfo 6488  ‘cfv 6490  (class class class)co 7356   qTop cqtop 17422  TopOnctopon 22852  KQckq 23635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-qtop 17426  df-top 22836  df-topon 22853  df-kq 23636

This theorem is referenced by:  kqt0lem  23678  isr0  23679  r0cld  23680  regr1lem2  23682  kqreglem1  23683  kqreglem2  23684  kqnrmlem1  23685  kqnrmlem2  23686  kqtop  23687