Theorem kqreg 21776

Description: The Kolmogorov quotient of a regular space is regular. By regr1 21775 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 21359	. . . 4 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top)
2		eqid 2760	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	toptopon 20944	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
4	1, 3	sylib 208	. . 3 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽))
5		eqid 2760	. . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
6	5	kqreglem1 21766	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)
7	4, 6	mpancom 706	. 2 ⊢ (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Reg)
8		regtop 21359	. . . . 5 ⊢ ((KQ‘𝐽) ∈ Reg → (KQ‘𝐽) ∈ Top)
9		kqtop 21770	. . . . 5 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
10	8, 9	sylibr 224	. . . 4 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Top)
11	10, 3	sylib 208	. . 3 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽))
12	5	kqreglem2 21767	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)
13	11, 12	mpancom 706	. 2 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Reg)
14	7, 13	impbii 199	1 ⊢ (𝐽 ∈ Reg ↔ (KQ‘𝐽) ∈ Reg)

Syntax hints:   ↔ wb 196   ∈ wcel 2139  {crab 3054  ∪ cuni 4588   ↦ cmpt 4881  ‘cfv 6049  Topctop 20920  TopOnctopon 20937  Regcreg 21335  KQckq 21718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-qtop 16389  df-top 20921  df-topon 20938  df-cld 21045  df-cls 21047  df-cn 21253  df-reg 21342  df-kq 21719

This theorem is referenced by: (None)