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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 22330 | . . . . . 6 β’ (π₯ β (topGenβπ΅) β π₯ β βͺ π΅) | |
2 | velpw 4566 | . . . . . 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 3949 | . . . 4 β’ (topGenβπ΅) β π« βͺ π΅ |
5 | sspwuni 5061 | . . . 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 22332 | . . 3 β’ (π΅ β π β π΅ β (topGenβπ΅)) | |
9 | 8 | unissd 4876 | . 2 β’ (π΅ β π β βͺ π΅ β βͺ (topGenβπ΅)) |
10 | 7, 9 | eqssd 3962 | 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 3911 π« cpw 4561 βͺ cuni 4866 βcfv 6497 topGenctg 17324 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-topgen 17330 |
This theorem is referenced by: tgcl 22335 tgtopon 22337 tgcmp 22768 2ndcsep 22826 txtopon 22958 ptuni 22961 xkouni 22966 prdstopn 22995 tgqtop 23079 alexsubb 23413 alexsubALTlem3 23416 alexsubALTlem4 23417 ptcmplem1 23419 uniretop 24142 fneval 34870 fnemeet1 34884 kelac2 41435 |
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