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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 22854 | . . . . . 6 β’ (π₯ β (topGenβπ΅) β π₯ β βͺ π΅) | |
| 2 | velpw 4603 | . . . . . 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 3982 | . . . 4 β’ (topGenβπ΅) β π« βͺ π΅ |
| 5 | sspwuni 5097 | . . . 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 22856 | . . 3 β’ (π΅ β π β π΅ β (topGenβπ΅)) | |
| 9 | 8 | unissd 4913 | . 2 β’ (π΅ β π β βͺ π΅ β βͺ (topGenβπ΅)) |
| 10 | 7, 9 | eqssd 3995 | 1 β’ (π΅ β π β βͺ (topGenβπ΅) = βͺ π΅) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1534 β wcel 2099 β wss 3944 π« cpw 4598 βͺ cuni 4903 βcfv 6542 topGenctg 17410 |
| 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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-topgen 17416 |
| This theorem is referenced by: tgcl 22859 tgtopon 22861 tgcmp 23292 2ndcsep 23350 txtopon 23482 ptuni 23485 xkouni 23490 prdstopn 23519 tgqtop 23603 alexsubb 23937 alexsubALTlem3 23940 alexsubALTlem4 23941 ptcmplem1 23943 uniretop 24666 fneval 35772 fnemeet1 35786 kelac2 42411 |
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