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Mirrors > Home > MPE Home > Th. List > t1r0 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
t1r0 | ⊢ (𝐽 ∈ Fre → (KQ‘𝐽) ∈ Fre) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | t1t0 21884 | . . 3 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2) | |
2 | kqhmph 22355 | . . 3 ⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽)) | |
3 | 1, 2 | sylib 219 | . 2 ⊢ (𝐽 ∈ Fre → 𝐽 ≃ (KQ‘𝐽)) |
4 | t1hmph 22327 | . 2 ⊢ (𝐽 ≃ (KQ‘𝐽) → (𝐽 ∈ Fre → (KQ‘𝐽) ∈ Fre)) | |
5 | 3, 4 | mpcom 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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