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Theorem t1r0 21845
Description: A T1 space is R0. That is, the Kolmogorov quotient of a T1 space is also T1 (because they are homeomorphic). (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
t1r0 (𝐽 ∈ Fre → (KQ‘𝐽) ∈ Fre)

Proof of Theorem t1r0
StepHypRef Expression
1 t1t0 21373 . . 3 (𝐽 ∈ Fre → 𝐽 ∈ Kol2)
2 kqhmph 21843 . . 3 (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))
31, 2sylib 208 . 2 (𝐽 ∈ Fre → 𝐽 ≃ (KQ‘𝐽))
4 t1hmph 21815 . 2 (𝐽 ≃ (KQ‘𝐽) → (𝐽 ∈ Fre → (KQ‘𝐽) ∈ Fre))
53, 4mpcom 38 1 (𝐽 ∈ Fre → (KQ‘𝐽) ∈ Fre)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145   class class class wbr 4786  cfv 6031  Kol2ct0 21331  Frect1 21332  KQckq 21717  chmph 21778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-1o 7713  df-map 8011  df-topgen 16312  df-qtop 16375  df-top 20919  df-topon 20936  df-cld 21044  df-cn 21252  df-t0 21338  df-t1 21339  df-kq 21718  df-hmeo 21779  df-hmph 21780
This theorem is referenced by:  nrmreg  21848
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