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| Mirrors > Home > MPE Home > Th. List > unitg | Structured version Visualization version GIF version | ||
| Description: The topology generated by a basis π΅ is a topology on βͺ π΅. Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class TopBases completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.) |
| Ref | Expression |
|---|---|
| unitg | β’ (π΅ β π β βͺ (topGenβπ΅) = βͺ π΅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tg1 22458 | . . . . . 6 β’ (π₯ β (topGenβπ΅) β π₯ β βͺ π΅) | |
| 2 | velpw 4606 | . . . . . 6 β’ (π₯ β π« βͺ π΅ β π₯ β βͺ π΅) | |
| 3 | 1, 2 | sylibr 233 | . . . . 5 β’ (π₯ β (topGenβπ΅) β π₯ β π« βͺ π΅) |
| 4 | 3 | ssriv 3985 | . . . 4 β’ (topGenβπ΅) β π« βͺ π΅ |
| 5 | sspwuni 5102 | . . . 4 β’ ((topGenβπ΅) β π« βͺ π΅ β βͺ (topGenβπ΅) β βͺ π΅) | |
| 6 | 4, 5 | mpbi 229 | . . 3 β’ βͺ (topGenβπ΅) β βͺ π΅ |
| 7 | 6 | a1i 11 | . 2 β’ (π΅ β π β βͺ (topGenβπ΅) β βͺ π΅) |
| 8 | bastg 22460 | . . 3 β’ (π΅ β π β π΅ β (topGenβπ΅)) | |
| 9 | 8 | unissd 4917 | . 2 β’ (π΅ β π β βͺ π΅ β βͺ (topGenβπ΅)) |
| 10 | 7, 9 | eqssd 3998 | 1 β’ (π΅ β π β βͺ (topGenβπ΅) = βͺ π΅) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1541 β wcel 2106 β wss 3947 π« cpw 4601 βͺ cuni 4907 βcfv 6540 topGenctg 17379 |
| 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-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-topgen 17385 |
| This theorem is referenced by: tgcl 22463 tgtopon 22465 tgcmp 22896 2ndcsep 22954 txtopon 23086 ptuni 23089 xkouni 23094 prdstopn 23123 tgqtop 23207 alexsubb 23541 alexsubALTlem3 23544 alexsubALTlem4 23545 ptcmplem1 23547 uniretop 24270 fneval 35225 fnemeet1 35239 kelac2 41792 |
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