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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 | β’ (π½ β Haus β (πGenβπ½) β Haus) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | haustop 23056 | . . . 4 β’ (π½ β Haus β π½ β Top) | |
| 2 | toptopon2 22641 | . . . 4 β’ (π½ β Top β π½ β (TopOnββͺ π½)) | |
| 3 | 1, 2 | sylib 217 | . . 3 β’ (π½ β Haus β π½ β (TopOnββͺ π½)) |
| 4 | kgentopon 23263 | . . 3 β’ (π½ β (TopOnββͺ π½) β (πGenβπ½) β (TopOnββͺ π½)) | |
| 5 | 3, 4 | syl 17 | . 2 β’ (π½ β Haus β (πGenβπ½) β (TopOnββͺ π½)) |
| 6 | kgenss 23268 | . . 3 β’ (π½ β Top β π½ β (πGenβπ½)) | |
| 7 | 1, 6 | syl 17 | . 2 β’ (π½ β Haus β π½ β (πGenβπ½)) |
| 8 | eqid 2731 | . . 3 β’ βͺ π½ = βͺ π½ | |
| 9 | 8 | sshaus 23100 | . 2 β’ ((π½ β Haus β§ (πGenβπ½) β (TopOnββͺ π½) β§ π½ β (πGenβπ½)) β (πGenβπ½) β Haus) |
| 10 | 5, 7, 9 | mpd3an23 1462 | 1 β’ (π½ β Haus β (πGenβπ½) β Haus) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wcel 2105 β wss 3948 βͺ cuni 4908 βcfv 6543 Topctop 22616 TopOnctopon 22633 Hauscha 23033 πGenckgen 23258 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 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 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-map 8826 df-en 8944 df-fin 8947 df-fi 9410 df-rest 17373 df-topgen 17394 df-top 22617 df-topon 22634 df-bases 22670 df-cn 22952 df-haus 23040 df-cmp 23112 df-kgen 23259 |
| This theorem is referenced by: (None) |
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