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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 12519 | . . . . . 6 ⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)}) | |
2 | 1 | funmpt2 5221 | . . . . 5 ⊢ Fun topGen |
3 | funrel 5199 | . . . . 5 ⊢ (Fun topGen → Rel topGen) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ Rel topGen |
5 | relelfvdm 5512 | . . . 4 ⊢ ((Rel topGen ∧ 𝐴 ∈ (topGen‘𝐵)) → 𝐵 ∈ dom topGen) | |
6 | 4, 5 | mpan 421 | . . 3 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen) |
7 | eltg2b 12601 | . . . 4 ⊢ (𝐵 ∈ dom topGen → (𝐴 ∈ (topGen‘𝐵) ↔ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ 𝐵 (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) | |
8 | eleq1 2227 | . . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝑥 ↔ 𝐶 ∈ 𝑥)) | |
9 | 8 | anbi1d 461 | . . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) ↔ (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
10 | 9 | rexbidv 2465 | . . . . 5 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐵 (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) ↔ ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
11 | 10 | rspccv 2822 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 ∈ 𝐵 (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → (𝐶 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
12 | 7, 11 | syl6bi 162 | . . 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 123 | 1 ⊢ ((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 {cab 2150 ∀wral 2442 ∃wrex 2443 Vcvv 2721 ∩ cin 3110 ⊆ wss 3111 𝒫 cpw 3553 ∪ cuni 3783 dom cdm 4598 Rel wrel 4603 Fun wfun 5176 ‘cfv 5182 topGenctg 12513 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-topgen 12519 |
This theorem is referenced by: tgclb 12612 tgcnp 12756 txlm 12826 |
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