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Mirrors > Home > MPE Home > Th. List > eltg2b | Structured version Visualization version GIF version |
Description: Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
eltg2b | ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eltg2 20573 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))) | |
2 | simpl 472 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) → 𝑥 ∈ 𝑦) | |
3 | 2 | reximi 2994 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) |
4 | eluni2 4376 | . . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) | |
5 | 3, 4 | sylibr 223 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) → 𝑥 ∈ ∪ 𝐵) |
6 | 5 | ralimi 2936 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) → ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵) |
7 | dfss3 3558 | . . . 4 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵) | |
8 | 6, 7 | sylibr 223 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) → 𝐴 ⊆ ∪ 𝐵) |
9 | 8 | pm4.71ri 663 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) |
10 | 1, 9 | syl6bbr 277 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ∪ cuni 4372 ‘cfv 5804 topGenctg 15921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-topgen 15927 |
This theorem is referenced by: tg2 20580 tgcl 20584 eltop2 20590 tgss2 20602 basgen2 20604 2ndc1stc 21064 eltx 21181 tgqioo 22411 isfne2 31507 |
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