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| Mirrors > Home > MPE Home > Th. List > tg2 | Structured version Visualization version 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 | elfvdm 6943 | . . 3 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen) | |
| 2 | eltg2b 22966 | . . . 4 ⊢ (𝐵 ∈ dom topGen → (𝐴 ∈ (topGen‘𝐵) ↔ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ 𝐵 (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) | |
| 3 | eleq1 2829 | . . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝑥 ↔ 𝐶 ∈ 𝑥)) | |
| 4 | 3 | anbi1d 631 | . . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) ↔ (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
| 5 | 4 | rexbidv 3179 | . . . . 5 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐵 (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) ↔ ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
| 6 | 5 | rspccv 3619 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 ∈ 𝐵 (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → (𝐶 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
| 7 | 2, 6 | biimtrdi 253 | . . 3 ⊢ (𝐵 ∈ dom topGen → (𝐴 ∈ (topGen‘𝐵) → (𝐶 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴)))) |
| 8 | 1, 7 | mpcom 38 | . 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → (𝐶 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
| 9 | 8 | imp 406 | 1 ⊢ ((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 dom cdm 5685 ‘cfv 6561 topGenctg 17482 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-topgen 17488 |
| This theorem is referenced by: tgclb 22977 elcls3 23091 pnfnei 23228 mnfnei 23229 tgcnp 23261 tgcmp 23409 2ndcctbss 23463 2ndcdisj 23464 2ndcomap 23466 dis2ndc 23468 ptpjopn 23620 txlm 23656 flftg 24004 alexsublem 24052 alexsubALT 24059 tmdgsum2 24104 xrge0tsms 24856 xrge0tsmsd 33065 iccllysconn 35255 rellysconn 35256 fnessex 36347 ptrecube 37627 islptre 45634 |
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