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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 22467 | . . . . . 6 β’ (π₯ β (topGenβπ΅) β π₯ β βͺ π΅) | |
| 2 | velpw 4608 | . . . . . 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 3987 | . . . 4 β’ (topGenβπ΅) β π« βͺ π΅ |
| 5 | sspwuni 5104 | . . . 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 22469 | . . 3 β’ (π΅ β π β π΅ β (topGenβπ΅)) | |
| 9 | 8 | unissd 4919 | . 2 β’ (π΅ β π β βͺ π΅ β βͺ (topGenβπ΅)) |
| 10 | 7, 9 | eqssd 4000 | 1 β’ (π΅ β π β βͺ (topGenβπ΅) = βͺ π΅) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3949 π« cpw 4603 βͺ cuni 4909 βcfv 6544 topGenctg 17383 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-topgen 17389 |
| This theorem is referenced by: tgcl 22472 tgtopon 22474 tgcmp 22905 2ndcsep 22963 txtopon 23095 ptuni 23098 xkouni 23103 prdstopn 23132 tgqtop 23216 alexsubb 23550 alexsubALTlem3 23553 alexsubALTlem4 23554 ptcmplem1 23556 uniretop 24279 fneval 35237 fnemeet1 35251 kelac2 41807 |
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