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Theorem kqreg 22354
Description: The Kolmogorov quotient of a regular space is regular. By regr1 22353 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.
StepHypRef Expression
1 regtop 21936 . . . 4 (𝐽 ∈ Reg → 𝐽 ∈ Top)
2 toptopon2 21521 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
31, 2sylib 221 . . 3 (𝐽 ∈ Reg → 𝐽 ∈ (TopOn‘ 𝐽))
4 eqid 2822 . . . 4 (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})
54kqreglem1 22344 . . 3 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)
63, 5mpancom 687 . 2 (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Reg)
7 regtop 21936 . . . . 5 ((KQ‘𝐽) ∈ Reg → (KQ‘𝐽) ∈ Top)
8 kqtop 22348 . . . . 5 (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
97, 8sylibr 237 . . . 4 ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Top)
109, 2sylib 221 . . 3 ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ (TopOn‘ 𝐽))
114kqreglem2 22345 . . 3 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)
1210, 11mpancom 687 . 2 ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Reg)
136, 12impbii 212 1 (𝐽 ∈ Reg ↔ (KQ‘𝐽) ∈ Reg)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2114  {crab 3134   cuni 4813  cmpt 5122  cfv 6334  Topctop 21496  TopOnctopon 21513  Regcreg 21912  KQckq 22296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-map 8395  df-qtop 16771  df-top 21497  df-topon 21514  df-cld 21622  df-cls 21624  df-cn 21830  df-reg 21919  df-kq 22297
This theorem is referenced by: (None)
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