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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 | β’ (π½ β Top β (πGenβπ½) β Top) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | toptopon2 22640 | . . 3 β’ (π½ β Top β π½ β (TopOnββͺ π½)) | |
| 2 | kgentopon 23262 | . . 3 β’ (π½ β (TopOnββͺ π½) β (πGenβπ½) β (TopOnββͺ π½)) | |
| 3 | 1, 2 | sylbi 216 | . 2 β’ (π½ β Top β (πGenβπ½) β (TopOnββͺ π½)) |
| 4 | topontop 22635 | . 2 β’ ((πGenβπ½) β (TopOnββͺ π½) β (πGenβπ½) β Top) | |
| 5 | 3, 4 | syl 17 | 1 β’ (π½ β Top β (πGenβπ½) β Top) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wcel 2106 βͺ cuni 4908 βcfv 6543 Topctop 22615 TopOnctopon 22632 πGenckgen 23257 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-en 8942 df-fin 8945 df-fi 9408 df-rest 17372 df-topgen 17393 df-top 22616 df-topon 22633 df-bases 22669 df-cmp 23111 df-kgen 23258 |
| This theorem is referenced by: kgenf 23265 kgencmp 23269 kgencmp2 23270 |
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