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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 22776 | . . 3 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2) | |
2 | kqhmph 23247 | . . 3 ⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽)) | |
3 | 1, 2 | sylib 217 | . 2 ⊢ (𝐽 ∈ Fre → 𝐽 ≃ (KQ‘𝐽)) |
4 | t1hmph 23219 | . 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 2106 class class class wbr 5138 ‘cfv 6529 Kol2ct0 22734 Frect1 22735 KQckq 23121 ≃ chmph 23182 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7954 df-2nd 7955 df-1o 8445 df-map 8802 df-topgen 17368 df-qtop 17432 df-top 22320 df-topon 22337 df-cld 22447 df-cn 22655 df-t0 22741 df-t1 22742 df-kq 23122 df-hmeo 23183 df-hmph 23184 |
This theorem is referenced by: nrmreg 23252 |
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