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Theorem t1r0 22357
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 21884 . . 3 (𝐽 ∈ Fre → 𝐽 ∈ Kol2)
2 kqhmph 22355 . . 3 (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))
31, 2sylib 219 . 2 (𝐽 ∈ Fre → 𝐽 ≃ (KQ‘𝐽))
4 t1hmph 22327 . 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 2105   class class class wbr 5057  cfv 6348  Kol2ct0 21842  Frect1 21843  KQckq 22229  chmph 22290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-1o 8091  df-map 8397  df-topgen 16705  df-qtop 16768  df-top 21430  df-topon 21447  df-cld 21555  df-cn 21763  df-t0 21849  df-t1 21850  df-kq 22230  df-hmeo 22291  df-hmph 22292
This theorem is referenced by:  nrmreg  22360
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