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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 21950 | . . 3 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2) | |
2 | kqhmph 22421 | . . 3 ⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽)) | |
3 | 1, 2 | sylib 220 | . 2 ⊢ (𝐽 ∈ Fre → 𝐽 ≃ (KQ‘𝐽)) |
4 | t1hmph 22393 | . 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 2110 class class class wbr 5058 ‘cfv 6349 Kol2ct0 21908 Frect1 21909 KQckq 22295 ≃ chmph 22356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-1o 8096 df-map 8402 df-topgen 16711 df-qtop 16774 df-top 21496 df-topon 21513 df-cld 21621 df-cn 21829 df-t0 21915 df-t1 21916 df-kq 22296 df-hmeo 22357 df-hmph 22358 |
This theorem is referenced by: nrmreg 22426 |
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