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| Mirrors > Home > ILE Home > Th. List > tgvalex | GIF version | ||
| Description: The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.) |
| Ref | Expression |
|---|---|
| tgvalex | β’ (π΅ β π β (topGenβπ΅) β V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tgval 12716 | . 2 β’ (π΅ β π β (topGenβπ΅) = {π¦ β£ π¦ β βͺ (π΅ β© π« π¦)}) | |
| 2 | inss1 3357 | . . . . . . 7 β’ (π΅ β© π« π¦) β π΅ | |
| 3 | 2 | unissi 3834 | . . . . . 6 β’ βͺ (π΅ β© π« π¦) β βͺ π΅ |
| 4 | sstr 3165 | . . . . . 6 β’ ((π¦ β βͺ (π΅ β© π« π¦) β§ βͺ (π΅ β© π« π¦) β βͺ π΅) β π¦ β βͺ π΅) | |
| 5 | 3, 4 | mpan2 425 | . . . . 5 β’ (π¦ β βͺ (π΅ β© π« π¦) β π¦ β βͺ π΅) |
| 6 | 5 | ss2abi 3229 | . . . 4 β’ {π¦ β£ π¦ β βͺ (π΅ β© π« π¦)} β {π¦ β£ π¦ β βͺ π΅} |
| 7 | df-pw 3579 | . . . 4 β’ π« βͺ π΅ = {π¦ β£ π¦ β βͺ π΅} | |
| 8 | 6, 7 | sseqtrri 3192 | . . 3 β’ {π¦ β£ π¦ β βͺ (π΅ β© π« π¦)} β π« βͺ π΅ |
| 9 | uniexg 4441 | . . . 4 β’ (π΅ β π β βͺ π΅ β V) | |
| 10 | 9 | pwexd 4183 | . . 3 β’ (π΅ β π β π« βͺ π΅ β V) |
| 11 | ssexg 4144 | . . 3 β’ (({π¦ β£ π¦ β βͺ (π΅ β© π« π¦)} β π« βͺ π΅ β§ π« βͺ π΅ β V) β {π¦ β£ π¦ β βͺ (π΅ β© π« π¦)} β V) | |
| 12 | 8, 10, 11 | sylancr 414 | . 2 β’ (π΅ β π β {π¦ β£ π¦ β βͺ (π΅ β© π« π¦)} β V) |
| 13 | 1, 12 | eqeltrd 2254 | 1 β’ (π΅ β π β (topGenβπ΅) β V) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β wcel 2148 {cab 2163 Vcvv 2739 β© cin 3130 β wss 3131 π« cpw 3577 βͺ cuni 3811 βcfv 5218 topGenctg 12708 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-topgen 12714 |
| This theorem is referenced by: ptex 12718 tgcl 13649 tgidm 13659 tgss3 13663 2basgeng 13667 tgrest 13754 txvalex 13839 txval 13840 txbasval 13852 |
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