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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 21643 | . . . 4 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top) | |
2 | toptopon2 21230 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
3 | 1, 2 | sylib 210 | . . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
4 | kgentopon 21850 | . . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽)) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐽 ∈ Haus → (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽)) |
6 | kgenss 21855 | . . 3 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽)) | |
7 | 1, 6 | syl 17 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ⊆ (𝑘Gen‘𝐽)) |
8 | eqid 2779 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
9 | 8 | sshaus 21687 | . 2 ⊢ ((𝐽 ∈ Haus ∧ (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ⊆ (𝑘Gen‘𝐽)) → (𝑘Gen‘𝐽) ∈ Haus) |
10 | 5, 7, 9 | mpd3an23 1442 | 1 ⊢ (𝐽 ∈ Haus → (𝑘Gen‘𝐽) ∈ Haus) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2050 ⊆ wss 3830 ∪ cuni 4712 ‘cfv 6188 Topctop 21205 TopOnctopon 21222 Hauscha 21620 𝑘Genckgen 21845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-oadd 7909 df-er 8089 df-map 8208 df-en 8307 df-fin 8310 df-fi 8670 df-rest 16552 df-topgen 16573 df-top 21206 df-topon 21223 df-bases 21258 df-cn 21539 df-haus 21627 df-cmp 21699 df-kgen 21846 |
This theorem is referenced by: (None) |
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