Theorem kqnrm 24083

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 23666	. . . 4 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2		toptopon2 23248	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3	1, 2	sylib 221	. . 3 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ (TopOn‘∪ 𝐽))
4		eqid 2763	. . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
5	4	kqnrmlem1 24074	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)
6	3, 5	mpancom 701	. 2 ⊢ (𝐽 ∈ Nrm → (KQ‘𝐽) ∈ Nrm)
7		nrmtop 23666	. . . . 5 ⊢ ((KQ‘𝐽) ∈ Nrm → (KQ‘𝐽) ∈ Top)
8		kqtop 24076	. . . . 5 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
9	7, 8	sylibr 237	. . . 4 ⊢ ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ Top)
10	9, 2	sylib 221	. . 3 ⊢ ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ (TopOn‘∪ 𝐽))
11	4	kqnrmlem2 24075	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)
12	10, 11	mpancom 701	. 2 ⊢ ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ Nrm)
13	6, 12	impbii 212	1 ⊢ (𝐽 ∈ Nrm ↔ (KQ‘𝐽) ∈ Nrm)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2146  {crab 3415  ∪ cuni 4869   ↦ cmpt 5188  ‘cfv 6546  Topctop 23223  TopOnctopon 23240  Nrmcnrm 23640  KQckq 24024

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-iin 4956  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  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 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-map 8860  df-qtop 17691  df-top 23224  df-topon 23241  df-cld 23349  df-cls 23351  df-cn 23557  df-nrm 23647  df-kq 24025

This theorem is used by: (None)