Theorem kqnrm 21884

Description: The Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)

Assertion

Ref	Expression
kqnrm	⊢ (𝐽 ∈ Nrm ↔ (KQ‘𝐽) ∈ Nrm)

Proof of Theorem kqnrm

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

Step	Hyp	Ref	Expression
1		nrmtop 21469	. . . 4 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2		eqid 2799	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	toptopon 21050	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
4	1, 3	sylib 210	. . 3 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ (TopOn‘∪ 𝐽))
5		eqid 2799	. . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
6	5	kqnrmlem1 21875	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)
7	4, 6	mpancom 680	. 2 ⊢ (𝐽 ∈ Nrm → (KQ‘𝐽) ∈ Nrm)
8		nrmtop 21469	. . . . 5 ⊢ ((KQ‘𝐽) ∈ Nrm → (KQ‘𝐽) ∈ Top)
9		kqtop 21877	. . . . 5 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
10	8, 9	sylibr 226	. . . 4 ⊢ ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ Top)
11	10, 3	sylib 210	. . 3 ⊢ ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ (TopOn‘∪ 𝐽))
12	5	kqnrmlem2 21876	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)
13	11, 12	mpancom 680	. 2 ⊢ ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ Nrm)
14	7, 13	impbii 201	1 ⊢ (𝐽 ∈ Nrm ↔ (KQ‘𝐽) ∈ Nrm)

Syntax hints:   ↔ wb 198   ∈ wcel 2157  {crab 3093  ∪ cuni 4628   ↦ cmpt 4922  ‘cfv 6101  Topctop 21026  TopOnctopon 21043  Nrmcnrm 21443  KQckq 21825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-iin 4713  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-qtop 16482  df-top 21027  df-topon 21044  df-cld 21152  df-cls 21154  df-cn 21360  df-nrm 21450  df-kq 21826

This theorem is referenced by: (None)