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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 | ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tgval 13065 | . 2 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)}) | |
| 2 | inss1 3392 | . . . . . . 7 ⊢ (𝐵 ∩ 𝒫 𝑦) ⊆ 𝐵 | |
| 3 | 2 | unissi 3872 | . . . . . 6 ⊢ ∪ (𝐵 ∩ 𝒫 𝑦) ⊆ ∪ 𝐵 |
| 4 | sstr 3200 | . . . . . 6 ⊢ ((𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦) ∧ ∪ (𝐵 ∩ 𝒫 𝑦) ⊆ ∪ 𝐵) → 𝑦 ⊆ ∪ 𝐵) | |
| 5 | 3, 4 | mpan2 425 | . . . . 5 ⊢ (𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦) → 𝑦 ⊆ ∪ 𝐵) |
| 6 | 5 | ss2abi 3264 | . . . 4 ⊢ {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ⊆ {𝑦 ∣ 𝑦 ⊆ ∪ 𝐵} |
| 7 | df-pw 3617 | . . . 4 ⊢ 𝒫 ∪ 𝐵 = {𝑦 ∣ 𝑦 ⊆ ∪ 𝐵} | |
| 8 | 6, 7 | sseqtrri 3227 | . . 3 ⊢ {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ⊆ 𝒫 ∪ 𝐵 |
| 9 | uniexg 4485 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V) | |
| 10 | 9 | pwexd 4224 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝒫 ∪ 𝐵 ∈ V) |
| 11 | ssexg 4182 | . . 3 ⊢ (({𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ⊆ 𝒫 ∪ 𝐵 ∧ 𝒫 ∪ 𝐵 ∈ V) → {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ∈ V) | |
| 12 | 8, 10, 11 | sylancr 414 | . 2 ⊢ (𝐵 ∈ 𝑉 → {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ∈ V) |
| 13 | 1, 12 | eqeltrd 2281 | 1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ∈ V) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∈ wcel 2175 {cab 2190 Vcvv 2771 ∩ cin 3164 ⊆ wss 3165 𝒫 cpw 3615 ∪ cuni 3849 ‘cfv 5270 topGenctg 13057 |
| 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 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-v 2773 df-sbc 2998 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-iota 5231 df-fun 5272 df-fv 5278 df-topgen 13063 |
| This theorem is referenced by: ptex 13067 mopnset 14285 tgcl 14507 tgidm 14517 tgss3 14521 2basgeng 14525 tgrest 14612 txvalex 14697 txval 14698 txbasval 14710 |
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