Theorem kqreg 23919

Description: The Kolmogorov quotient of a regular space is regular. By regr1 23918 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 23501	. . . 4 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top)
2		toptopon2 23086	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3	1, 2	sylib 221	. . 3 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽))
4		eqid 2762	. . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
5	4	kqreglem1 23909	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)
6	3, 5	mpancom 700	. 2 ⊢ (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Reg)
7		regtop 23501	. . . . 5 ⊢ ((KQ‘𝐽) ∈ Reg → (KQ‘𝐽) ∈ Top)
8		kqtop 23913	. . . . 5 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
9	7, 8	sylibr 237	. . . 4 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Top)
10	9, 2	sylib 221	. . 3 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽))
11	4	kqreglem2 23910	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)
12	10, 11	mpancom 700	. 2 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Reg)
13	6, 12	impbii 212	1 ⊢ (𝐽 ∈ Reg ↔ (KQ‘𝐽) ∈ Reg)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2142  {crab 3415  ∪ cuni 4871   ↦ cmpt 5191  ‘cfv 6536  Topctop 23061  TopOnctopon 23078  Regcreg 23477  KQckq 23861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8824  df-qtop 17567  df-top 23062  df-topon 23079  df-cld 23187  df-cls 23189  df-cn 23395  df-reg 23484  df-kq 23862

This theorem is used by: (None)