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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 20945 | . . . 4 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top) | |
2 | eqid 2610 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | 2 | toptopon 20548 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
4 | 1, 3 | sylib 207 | . . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
5 | kgentopon 21151 | . . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐽 ∈ Haus → (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽)) |
7 | kgenss 21156 | . . 3 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽)) | |
8 | 1, 7 | syl 17 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ⊆ (𝑘Gen‘𝐽)) |
9 | 2 | sshaus 20989 | . 2 ⊢ ((𝐽 ∈ Haus ∧ (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ⊆ (𝑘Gen‘𝐽)) → (𝑘Gen‘𝐽) ∈ Haus) |
10 | 6, 8, 9 | mpd3an23 1418 | 1 ⊢ (𝐽 ∈ Haus → (𝑘Gen‘𝐽) ∈ Haus) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 ‘cfv 5804 Topctop 20517 TopOnctopon 20518 Hauscha 20922 𝑘Genckgen 21146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-fin 7845 df-fi 8200 df-rest 15906 df-topgen 15927 df-top 20521 df-bases 20522 df-topon 20523 df-cn 20841 df-haus 20929 df-cmp 21000 df-kgen 21147 |
This theorem is referenced by: (None) |
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