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Mirrors > Home > ILE Home > Th. List > eltg4i | 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 | df-topgen 12697 | . . . . . . 7 ⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)}) | |
2 | 1 | funmpt2 5254 | . . . . . 6 ⊢ Fun topGen |
3 | funrel 5232 | . . . . . 6 ⊢ (Fun topGen → Rel topGen) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ Rel topGen |
5 | relelfvdm 5546 | . . . . 5 ⊢ ((Rel topGen ∧ 𝐴 ∈ (topGen‘𝐵)) → 𝐵 ∈ dom topGen) | |
6 | 4, 5 | mpan 424 | . . . 4 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen) |
7 | eltg 13423 | . . . 4 ⊢ (𝐵 ∈ dom topGen → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴))) | |
8 | 6, 7 | syl 14 | . . 3 ⊢ (𝐴 ∈ (topGen‘𝐵) → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴))) |
9 | 8 | ibi 176 | . 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)) |
10 | inss2 3356 | . . . . 5 ⊢ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴 | |
11 | 10 | unissi 3832 | . . . 4 ⊢ ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ ∪ 𝒫 𝐴 |
12 | unipw 4216 | . . . 4 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
13 | 11, 12 | sseqtri 3189 | . . 3 ⊢ ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴 |
14 | 13 | a1i 9 | . 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴) |
15 | 9, 14 | eqssd 3172 | 1 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 = ∪ (𝐵 ∩ 𝒫 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 {cab 2163 Vcvv 2737 ∩ cin 3128 ⊆ wss 3129 𝒫 cpw 3575 ∪ cuni 3809 dom cdm 4625 Rel wrel 4630 Fun wfun 5209 ‘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: eltg3 13428 tgdom 13443 tgidm 13445 |
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