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Mirrors > Home > ILE Home > Th. List > tg2 | GIF version |
Description: Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.) |
Ref | Expression |
---|---|
tg2 | ⊢ ((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-topgen 12762 | . . . . . 6 ⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)}) | |
2 | 1 | funmpt2 5274 | . . . . 5 ⊢ Fun topGen |
3 | funrel 5252 | . . . . 5 ⊢ (Fun topGen → Rel topGen) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ Rel topGen |
5 | relelfvdm 5566 | . . . 4 ⊢ ((Rel topGen ∧ 𝐴 ∈ (topGen‘𝐵)) → 𝐵 ∈ dom topGen) | |
6 | 4, 5 | mpan 424 | . . 3 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen) |
7 | eltg2b 14006 | . . . 4 ⊢ (𝐵 ∈ dom topGen → (𝐴 ∈ (topGen‘𝐵) ↔ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ 𝐵 (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) | |
8 | eleq1 2252 | . . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝑥 ↔ 𝐶 ∈ 𝑥)) | |
9 | 8 | anbi1d 465 | . . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) ↔ (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
10 | 9 | rexbidv 2491 | . . . . 5 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐵 (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) ↔ ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
11 | 10 | rspccv 2853 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 ∈ 𝐵 (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → (𝐶 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
12 | 7, 11 | biimtrdi 163 | . . 3 ⊢ (𝐵 ∈ dom topGen → (𝐴 ∈ (topGen‘𝐵) → (𝐶 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴)))) |
13 | 6, 12 | mpcom 36 | . 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → (𝐶 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
14 | 13 | imp 124 | 1 ⊢ ((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 {cab 2175 ∀wral 2468 ∃wrex 2469 Vcvv 2752 ∩ cin 3143 ⊆ wss 3144 𝒫 cpw 3590 ∪ cuni 3824 dom cdm 4644 Rel wrel 4649 Fun wfun 5229 ‘cfv 5235 topGenctg 12756 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-topgen 12762 |
This theorem is referenced by: tgclb 14017 tgcnp 14161 txlm 14231 |
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