Theorem kqnrm 23672

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 23256	. . . 4 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2		toptopon2 22838	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3	1, 2	sylib 218	. . 3 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ (TopOn‘∪ 𝐽))
4		eqid 2729	. . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
5	4	kqnrmlem1 23663	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)
6	3, 5	mpancom 688	. 2 ⊢ (𝐽 ∈ Nrm → (KQ‘𝐽) ∈ Nrm)
7		nrmtop 23256	. . . . 5 ⊢ ((KQ‘𝐽) ∈ Nrm → (KQ‘𝐽) ∈ Top)
8		kqtop 23665	. . . . 5 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
9	7, 8	sylibr 234	. . . 4 ⊢ ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ Top)
10	9, 2	sylib 218	. . 3 ⊢ ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ (TopOn‘∪ 𝐽))
11	4	kqnrmlem2 23664	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)
12	10, 11	mpancom 688	. 2 ⊢ ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ Nrm)
13	6, 12	impbii 209	1 ⊢ (𝐽 ∈ Nrm ↔ (KQ‘𝐽) ∈ Nrm)

Syntax hints:   ↔ wb 206   ∈ wcel 2109  {crab 3402  ∪ cuni 4867   ↦ cmpt 5183  ‘cfv 6499  Topctop 22813  TopOnctopon 22830  Nrmcnrm 23230  KQckq 23613

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8778  df-qtop 17446  df-top 22814  df-topon 22831  df-cld 22939  df-cls 22941  df-cn 23147  df-nrm 23237  df-kq 23614

This theorem is referenced by: (None)