Theorem kqreg 22353

Description: The Kolmogorov quotient of a regular space is regular. By regr1 22352 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 21935	. . . 4 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top)
2		toptopon2 21520	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3	1, 2	sylib 220	. . 3 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽))
4		eqid 2821	. . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
5	4	kqreglem1 22343	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)
6	3, 5	mpancom 686	. 2 ⊢ (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Reg)
7		regtop 21935	. . . . 5 ⊢ ((KQ‘𝐽) ∈ Reg → (KQ‘𝐽) ∈ Top)
8		kqtop 22347	. . . . 5 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
9	7, 8	sylibr 236	. . . 4 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Top)
10	9, 2	sylib 220	. . 3 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽))
11	4	kqreglem2 22344	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)
12	10, 11	mpancom 686	. 2 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Reg)
13	6, 12	impbii 211	1 ⊢ (𝐽 ∈ Reg ↔ (KQ‘𝐽) ∈ Reg)

Syntax hints:   ↔ wb 208   ∈ wcel 2110  {crab 3142  ∪ cuni 4832   ↦ cmpt 5139  ‘cfv 6350  Topctop 21495  TopOnctopon 21512  Regcreg 21911  KQckq 22295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8402  df-qtop 16774  df-top 21496  df-topon 21513  df-cld 21621  df-cls 21623  df-cn 21829  df-reg 21918  df-kq 22296

This theorem is referenced by: (None)