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Mirrors > Home > MPE Home > Th. List > kgenhaus | Structured version Visualization version GIF version |
Description: The compact generator generates another Hausdorff topology given a Hausdorff topology to start from. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
kgenhaus | ⊢ (𝐽 ∈ Haus → (𝑘Gen‘𝐽) ∈ Haus) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | haustop 22492 | . . . 4 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top) | |
2 | toptopon2 22077 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
3 | 1, 2 | sylib 217 | . . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
4 | kgentopon 22699 | . . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽)) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐽 ∈ Haus → (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽)) |
6 | kgenss 22704 | . . 3 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽)) | |
7 | 1, 6 | syl 17 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ⊆ (𝑘Gen‘𝐽)) |
8 | eqid 2738 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
9 | 8 | sshaus 22536 | . 2 ⊢ ((𝐽 ∈ Haus ∧ (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ⊆ (𝑘Gen‘𝐽)) → (𝑘Gen‘𝐽) ∈ Haus) |
10 | 5, 7, 9 | mpd3an23 1462 | 1 ⊢ (𝐽 ∈ Haus → (𝑘Gen‘𝐽) ∈ Haus) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ⊆ wss 3886 ∪ cuni 4839 ‘cfv 6426 Topctop 22052 TopOnctopon 22069 Hauscha 22469 𝑘Genckgen 22694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-map 8604 df-en 8721 df-fin 8724 df-fi 9157 df-rest 17143 df-topgen 17164 df-top 22053 df-topon 22070 df-bases 22106 df-cn 22388 df-haus 22476 df-cmp 22548 df-kgen 22695 |
This theorem is referenced by: (None) |
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