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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 13420 | . 2 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)}) | |
2 | inss1 3355 | . . . . . . 7 ⊢ (𝐵 ∩ 𝒫 𝑦) ⊆ 𝐵 | |
3 | 2 | unissi 3832 | . . . . . 6 ⊢ ∪ (𝐵 ∩ 𝒫 𝑦) ⊆ ∪ 𝐵 |
4 | sstr 3163 | . . . . . 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 3227 | . . . 4 ⊢ {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ⊆ {𝑦 ∣ 𝑦 ⊆ ∪ 𝐵} |
7 | df-pw 3577 | . . . 4 ⊢ 𝒫 ∪ 𝐵 = {𝑦 ∣ 𝑦 ⊆ ∪ 𝐵} | |
8 | 6, 7 | sseqtrri 3190 | . . 3 ⊢ {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ⊆ 𝒫 ∪ 𝐵 |
9 | uniexg 4438 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V) | |
10 | 9 | pwexd 4180 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝒫 ∪ 𝐵 ∈ V) |
11 | ssexg 4141 | . . 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 2254 | 1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 {cab 2163 Vcvv 2737 ∩ cin 3128 ⊆ wss 3129 𝒫 cpw 3575 ∪ cuni 3809 ‘cfv 5215 topGenctg 12691 |
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 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 |
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 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5177 df-fun 5217 df-fv 5223 df-topgen 12697 |
This theorem is referenced by: tgcl 13435 tgidm 13445 tgss3 13449 2basgeng 13453 tgrest 13540 txvalex 13625 txval 13626 txbasval 13638 |
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