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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 12068 | . . . . . . 7 ⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)}) | |
2 | 1 | funmpt2 5132 | . . . . . 6 ⊢ Fun topGen |
3 | funrel 5110 | . . . . . 6 ⊢ (Fun topGen → Rel topGen) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ Rel topGen |
5 | relelfvdm 5421 | . . . . 5 ⊢ ((Rel topGen ∧ 𝐴 ∈ (topGen‘𝐵)) → 𝐵 ∈ dom topGen) | |
6 | 4, 5 | mpan 420 | . . . 4 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen) |
7 | eltg 12148 | . . . 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 175 | . 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)) |
10 | inss2 3267 | . . . . 5 ⊢ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴 | |
11 | 10 | unissi 3729 | . . . 4 ⊢ ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ ∪ 𝒫 𝐴 |
12 | unipw 4109 | . . . 4 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
13 | 11, 12 | sseqtri 3101 | . . 3 ⊢ ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴 |
14 | 13 | a1i 9 | . 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴) |
15 | 9, 14 | eqssd 3084 | 1 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 = ∪ (𝐵 ∩ 𝒫 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∈ wcel 1465 {cab 2103 Vcvv 2660 ∩ cin 3040 ⊆ wss 3041 𝒫 cpw 3480 ∪ cuni 3706 dom cdm 4509 Rel wrel 4514 Fun wfun 5087 ‘cfv 5093 topGenctg 12062 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-topgen 12068 |
This theorem is referenced by: eltg3 12153 tgdom 12168 tgidm 12170 |
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