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Mirrors > Home > MPE Home > Th. List > eltg2 | Structured version Visualization version GIF version |
Description: Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
eltg2 | ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgval2 21492 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))}) | |
2 | 1 | eleq2d 2895 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))})) |
3 | elex 3510 | . . . 4 ⊢ (𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))} → 𝐴 ∈ V) | |
4 | 3 | adantl 482 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))}) → 𝐴 ∈ V) |
5 | uniexg 7456 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V) | |
6 | ssexg 5218 | . . . . . 6 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ V) → 𝐴 ∈ V) | |
7 | 5, 6 | sylan2 592 | . . . . 5 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V) |
8 | 7 | ancoms 459 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝐵) → 𝐴 ∈ V) |
9 | 8 | adantrr 713 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) → 𝐴 ∈ V) |
10 | sseq1 3989 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ 𝐵)) | |
11 | sseq2 3990 | . . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝐴)) | |
12 | 11 | anbi2d 628 | . . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) |
13 | 12 | rexbidv 3294 | . . . . . 6 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧) ↔ ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) |
14 | 13 | raleqbi1dv 3401 | . . . . 5 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) |
15 | 10, 14 | anbi12d 630 | . . . 4 ⊢ (𝑧 = 𝐴 → ((𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧)) ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))) |
16 | 15 | elabg 3663 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))} ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))) |
17 | 4, 9, 16 | pm5.21nd 798 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))} ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))) |
18 | 2, 17 | bitrd 280 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {cab 2796 ∀wral 3135 ∃wrex 3136 Vcvv 3492 ⊆ wss 3933 ∪ cuni 4830 ‘cfv 6348 topGenctg 16699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-topgen 16705 |
This theorem is referenced by: eltg2b 21495 tg1 21500 tgcl 21505 elmopn 22979 psmetutop 23104 |
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