![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 21284 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))}) | |
2 | 1 | eleq2d 2846 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))})) |
3 | elex 3428 | . . . 4 ⊢ (𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))} → 𝐴 ∈ V) | |
4 | 3 | adantl 474 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))}) → 𝐴 ∈ V) |
5 | uniexg 7284 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V) | |
6 | ssexg 5080 | . . . . . 6 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ V) → 𝐴 ∈ V) | |
7 | 5, 6 | sylan2 584 | . . . . 5 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V) |
8 | 7 | ancoms 451 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝐵) → 𝐴 ∈ V) |
9 | 8 | adantrr 705 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) → 𝐴 ∈ V) |
10 | sseq1 3877 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ 𝐵)) | |
11 | sseq2 3878 | . . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝐴)) | |
12 | 11 | anbi2d 620 | . . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) |
13 | 12 | rexbidv 3237 | . . . . . 6 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧) ↔ ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) |
14 | 13 | raleqbi1dv 3338 | . . . . 5 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) |
15 | 10, 14 | anbi12d 622 | . . . 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 3575 | . . 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 790 | . 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 271 | 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 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 {cab 2753 ∀wral 3083 ∃wrex 3084 Vcvv 3410 ⊆ wss 3824 ∪ cuni 4709 ‘cfv 6186 topGenctg 16566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-sbc 3677 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-iota 6150 df-fun 6188 df-fv 6194 df-topgen 16572 |
This theorem is referenced by: eltg2b 21287 tg1 21292 tgcl 21297 elmopn 22771 psmetutop 22896 |
Copyright terms: Public domain | W3C validator |