Theorem kqreg 21932

Description: The Kolmogorov quotient of a regular space is regular. By regr1 21931 it is also Hausdorff, so we can also say that a space is regular iff the Kolmogorov quotient is regular Hausdorff (T₃). (Contributed by Mario Carneiro, 25-Aug-2015.)

Assertion

Ref	Expression
kqreg	⊢ (𝐽 ∈ Reg ↔ (KQ‘𝐽) ∈ Reg)

Proof of Theorem kqreg

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

Step	Hyp	Ref	Expression
1		regtop 21515	. . . 4 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top)
2		eqid 2825	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	toptopon 21099	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
4	1, 3	sylib 210	. . 3 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽))
5		eqid 2825	. . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
6	5	kqreglem1 21922	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)
7	4, 6	mpancom 679	. 2 ⊢ (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Reg)
8		regtop 21515	. . . . 5 ⊢ ((KQ‘𝐽) ∈ Reg → (KQ‘𝐽) ∈ Top)
9		kqtop 21926	. . . . 5 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
10	8, 9	sylibr 226	. . . 4 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Top)
11	10, 3	sylib 210	. . 3 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽))
12	5	kqreglem2 21923	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)
13	11, 12	mpancom 679	. 2 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Reg)
14	7, 13	impbii 201	1 ⊢ (𝐽 ∈ Reg ↔ (KQ‘𝐽) ∈ Reg)

Syntax hints:   ↔ wb 198   ∈ wcel 2164  {crab 3121  ∪ cuni 4660   ↦ cmpt 4954  ‘cfv 6127  Topctop 21075  TopOnctopon 21092  Regcreg 21491  KQckq 21874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-map 8129  df-qtop 16527  df-top 21076  df-topon 21093  df-cld 21201  df-cls 21203  df-cn 21409  df-reg 21498  df-kq 21875

This theorem is referenced by: (None)