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Mirrors > Home > ILE Home > Th. List > eltg3 | GIF version |
Description: Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.) |
Ref | Expression |
---|---|
eltg3 | ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 = ∪ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-topgen 11825 | . . . . . . 7 ⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)}) | |
2 | 1 | funmpt2 5087 | . . . . . 6 ⊢ Fun topGen |
3 | funrel 5066 | . . . . . 6 ⊢ (Fun topGen → Rel topGen) | |
4 | 2, 3 | ax-mp 7 | . . . . 5 ⊢ Rel topGen |
5 | relelfvdm 5371 | . . . . 5 ⊢ ((Rel topGen ∧ 𝐴 ∈ (topGen‘𝐵)) → 𝐵 ∈ dom topGen) | |
6 | 4, 5 | mpan 416 | . . . 4 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen) |
7 | inex1g 3996 | . . . 4 ⊢ (𝐵 ∈ dom topGen → (𝐵 ∩ 𝒫 𝐴) ∈ V) | |
8 | 6, 7 | syl 14 | . . 3 ⊢ (𝐴 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝐴) ∈ V) |
9 | eltg4i 11907 | . . 3 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 = ∪ (𝐵 ∩ 𝒫 𝐴)) | |
10 | inss1 3235 | . . . . . . 7 ⊢ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐵 | |
11 | sseq1 3062 | . . . . . . 7 ⊢ (𝑥 = (𝐵 ∩ 𝒫 𝐴) → (𝑥 ⊆ 𝐵 ↔ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐵)) | |
12 | 10, 11 | mpbiri 167 | . . . . . 6 ⊢ (𝑥 = (𝐵 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝐵) |
13 | 12 | biantrurd 300 | . . . . 5 ⊢ (𝑥 = (𝐵 ∩ 𝒫 𝐴) → (𝐴 = ∪ 𝑥 ↔ (𝑥 ⊆ 𝐵 ∧ 𝐴 = ∪ 𝑥))) |
14 | unieq 3684 | . . . . . 6 ⊢ (𝑥 = (𝐵 ∩ 𝒫 𝐴) → ∪ 𝑥 = ∪ (𝐵 ∩ 𝒫 𝐴)) | |
15 | 14 | eqeq2d 2106 | . . . . 5 ⊢ (𝑥 = (𝐵 ∩ 𝒫 𝐴) → (𝐴 = ∪ 𝑥 ↔ 𝐴 = ∪ (𝐵 ∩ 𝒫 𝐴))) |
16 | 13, 15 | bitr3d 189 | . . . 4 ⊢ (𝑥 = (𝐵 ∩ 𝒫 𝐴) → ((𝑥 ⊆ 𝐵 ∧ 𝐴 = ∪ 𝑥) ↔ 𝐴 = ∪ (𝐵 ∩ 𝒫 𝐴))) |
17 | 16 | spcegv 2721 | . . 3 ⊢ ((𝐵 ∩ 𝒫 𝐴) ∈ V → (𝐴 = ∪ (𝐵 ∩ 𝒫 𝐴) → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 = ∪ 𝑥))) |
18 | 8, 9, 17 | sylc 62 | . 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 = ∪ 𝑥)) |
19 | eltg3i 11908 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ⊆ 𝐵) → ∪ 𝑥 ∈ (topGen‘𝐵)) | |
20 | eleq1 2157 | . . . . 5 ⊢ (𝐴 = ∪ 𝑥 → (𝐴 ∈ (topGen‘𝐵) ↔ ∪ 𝑥 ∈ (topGen‘𝐵))) | |
21 | 19, 20 | syl5ibrcom 156 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ⊆ 𝐵) → (𝐴 = ∪ 𝑥 → 𝐴 ∈ (topGen‘𝐵))) |
22 | 21 | expimpd 356 | . . 3 ⊢ (𝐵 ∈ 𝑉 → ((𝑥 ⊆ 𝐵 ∧ 𝐴 = ∪ 𝑥) → 𝐴 ∈ (topGen‘𝐵))) |
23 | 22 | exlimdv 1754 | . 2 ⊢ (𝐵 ∈ 𝑉 → (∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 = ∪ 𝑥) → 𝐴 ∈ (topGen‘𝐵))) |
24 | 18, 23 | impbid2 142 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 = ∪ 𝑥))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1296 ∃wex 1433 ∈ wcel 1445 {cab 2081 Vcvv 2633 ∩ cin 3012 ⊆ wss 3013 𝒫 cpw 3449 ∪ cuni 3675 dom cdm 4467 Rel wrel 4472 Fun wfun 5043 ‘cfv 5049 topGenctg 11819 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-topgen 11825 |
This theorem is referenced by: tgval3 11910 tgtop 11920 eltop3 11923 tgidm 11926 bastop1 11935 tgrest 12021 tgcn 12059 |
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