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Mirrors > Home > MPE Home > Th. List > kgenftop | Structured version Visualization version GIF version |
Description: The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015.) |
Ref | Expression |
---|---|
kgenftop | ⊢ (𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toptopon2 22908 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
2 | kgentopon 23530 | . . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽)) | |
3 | 1, 2 | sylbi 216 | . 2 ⊢ (𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽)) |
4 | topontop 22903 | . 2 ⊢ ((𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽) → (𝑘Gen‘𝐽) ∈ Top) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ∪ cuni 4905 ‘cfv 6546 Topctop 22883 TopOnctopon 22900 𝑘Genckgen 23525 |
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 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-en 8967 df-fin 8970 df-fi 9447 df-rest 17432 df-topgen 17453 df-top 22884 df-topon 22901 df-bases 22937 df-cmp 23379 df-kgen 23526 |
This theorem is referenced by: kgenf 23533 kgencmp 23537 kgencmp2 23538 |
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