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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 | ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 = ∪ (𝐵 ∩ 𝒫 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6677 | . . . 4 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen) | |
2 | eltg 21562 | . . . 4 ⊢ (𝐵 ∈ dom topGen → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ∈ (topGen‘𝐵) → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴))) |
4 | 3 | ibi 270 | . 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)) |
5 | inss2 4156 | . . . . 5 ⊢ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴 | |
6 | 5 | unissi 4809 | . . . 4 ⊢ ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ ∪ 𝒫 𝐴 |
7 | unipw 5308 | . . . 4 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
8 | 6, 7 | sseqtri 3951 | . . 3 ⊢ ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴 |
9 | 8 | a1i 11 | . 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴) |
10 | 4, 9 | eqssd 3932 | 1 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 = ∪ (𝐵 ∩ 𝒫 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 dom cdm 5519 ‘cfv 6324 topGenctg 16703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-topgen 16709 |
This theorem is referenced by: eltg3 21567 tgdom 21583 tgidm 21585 ontgval 33892 |
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