Theorem kqreg 23780

Description: The Kolmogorov quotient of a regular space is regular. By regr1 23779 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 23362	. . . 4 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top)
2		toptopon2 22947	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3	1, 2	sylib 220	. . 3 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽))
4		eqid 2752	. . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
5	4	kqreglem1 23770	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)
6	3, 5	mpancom 696	. 2 ⊢ (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Reg)
7		regtop 23362	. . . . 5 ⊢ ((KQ‘𝐽) ∈ Reg → (KQ‘𝐽) ∈ Top)
8		kqtop 23774	. . . . 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 23771	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)
12	10, 11	mpancom 696	. 2 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Reg)
13	6, 12	impbii 211	1 ⊢ (𝐽 ∈ Reg ↔ (KQ‘𝐽) ∈ Reg)

Syntax hints:   ↔ wb 208   ∈ wcel 2132  {crab 3404  ∪ cuni 4855   ↦ cmpt 5171  ‘cfv 6506  Topctop 22922  TopOnctopon 22939  Regcreg 23338  KQckq 23722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-iin 4942  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-map 8794  df-qtop 17509  df-top 22923  df-topon 22940  df-cld 23048  df-cls 23050  df-cn 23256  df-reg 23345  df-kq 23723

This theorem is referenced by: (None)