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Mirrors > Home > MPE Home > Th. List > kgenuni | Structured version Visualization version GIF version |
Description: The base set of the compact generator is the same as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.) |
Ref | Expression |
---|---|
kgenuni.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
kgenuni | ⊢ (𝐽 ∈ Top → 𝑋 = ∪ (𝑘Gen‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kgenuni.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | toptopon 22910 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
3 | kgentopon 23533 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋)) | |
4 | 2, 3 | sylbi 216 | . 2 ⊢ (𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋)) |
5 | toponuni 22907 | . 2 ⊢ ((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) → 𝑋 = ∪ (𝑘Gen‘𝐽)) | |
6 | 4, 5 | syl 17 | 1 ⊢ (𝐽 ∈ Top → 𝑋 = ∪ (𝑘Gen‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∪ cuni 4913 ‘cfv 6554 Topctop 22886 TopOnctopon 22903 𝑘Genckgen 23528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-en 8975 df-fin 8978 df-fi 9454 df-rest 17437 df-topgen 17458 df-top 22887 df-topon 22904 df-bases 22940 df-cmp 23382 df-kgen 23529 |
This theorem is referenced by: kgencmp2 23541 llycmpkgen2 23545 1stckgen 23549 txkgen 23647 qtopkgen 23705 |
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