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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 12590 | . 2 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)}) | |
2 | inss1 3337 | . . . . . . 7 ⊢ (𝐵 ∩ 𝒫 𝑦) ⊆ 𝐵 | |
3 | 2 | unissi 3806 | . . . . . 6 ⊢ ∪ (𝐵 ∩ 𝒫 𝑦) ⊆ ∪ 𝐵 |
4 | sstr 3145 | . . . . . 6 ⊢ ((𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦) ∧ ∪ (𝐵 ∩ 𝒫 𝑦) ⊆ ∪ 𝐵) → 𝑦 ⊆ ∪ 𝐵) | |
5 | 3, 4 | mpan2 422 | . . . . 5 ⊢ (𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦) → 𝑦 ⊆ ∪ 𝐵) |
6 | 5 | ss2abi 3209 | . . . 4 ⊢ {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ⊆ {𝑦 ∣ 𝑦 ⊆ ∪ 𝐵} |
7 | df-pw 3555 | . . . 4 ⊢ 𝒫 ∪ 𝐵 = {𝑦 ∣ 𝑦 ⊆ ∪ 𝐵} | |
8 | 6, 7 | sseqtrri 3172 | . . 3 ⊢ {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ⊆ 𝒫 ∪ 𝐵 |
9 | uniexg 4411 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V) | |
10 | 9 | pwexd 4154 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝒫 ∪ 𝐵 ∈ V) |
11 | ssexg 4115 | . . 3 ⊢ (({𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ⊆ 𝒫 ∪ 𝐵 ∧ 𝒫 ∪ 𝐵 ∈ V) → {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ∈ V) | |
12 | 8, 10, 11 | sylancr 411 | . 2 ⊢ (𝐵 ∈ 𝑉 → {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ∈ V) |
13 | 1, 12 | eqeltrd 2241 | 1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 {cab 2150 Vcvv 2721 ∩ cin 3110 ⊆ wss 3111 𝒫 cpw 3553 ∪ cuni 3783 ‘cfv 5182 topGenctg 12507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-topgen 12513 |
This theorem is referenced by: tgcl 12605 tgidm 12615 tgss3 12619 2basgeng 12623 tgrest 12710 txvalex 12795 txval 12796 txbasval 12808 |
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