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Mirrors > Home > MPE Home > Th. List > eltg4i | Structured version Visualization version GIF version |
Description: An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
eltg4i | β’ (π΄ β (topGenβπ΅) β π΄ = βͺ (π΅ β© π« π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6880 | . . . 4 β’ (π΄ β (topGenβπ΅) β π΅ β dom topGen) | |
2 | eltg 22323 | . . . 4 β’ (π΅ β dom topGen β (π΄ β (topGenβπ΅) β π΄ β βͺ (π΅ β© π« π΄))) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π΄ β (topGenβπ΅) β (π΄ β (topGenβπ΅) β π΄ β βͺ (π΅ β© π« π΄))) |
4 | 3 | ibi 267 | . 2 β’ (π΄ β (topGenβπ΅) β π΄ β βͺ (π΅ β© π« π΄)) |
5 | inss2 4190 | . . . . 5 β’ (π΅ β© π« π΄) β π« π΄ | |
6 | 5 | unissi 4875 | . . . 4 β’ βͺ (π΅ β© π« π΄) β βͺ π« π΄ |
7 | unipw 5408 | . . . 4 β’ βͺ π« π΄ = π΄ | |
8 | 6, 7 | sseqtri 3981 | . . 3 β’ βͺ (π΅ β© π« π΄) β π΄ |
9 | 8 | a1i 11 | . 2 β’ (π΄ β (topGenβπ΅) β βͺ (π΅ β© π« π΄) β π΄) |
10 | 4, 9 | eqssd 3962 | 1 β’ (π΄ β (topGenβπ΅) β π΄ = βͺ (π΅ β© π« π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 β© cin 3910 β wss 3911 π« cpw 4561 βͺ cuni 4866 dom cdm 5634 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: eltg3 22328 tgdom 22344 tgidm 22346 ontgval 34949 |
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