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Theorem tgval2 21602
 Description: Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 21615) that (topGen‘𝐵) is indeed a topology (on ∪ 𝐵, see unitg 21613). See also tgval 21601 and tgval3 21609. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
Assertion
Ref Expression
tgval2 (𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))})
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝑉,𝑦,𝑧

Proof of Theorem tgval2
StepHypRef Expression
1 tgval 21601 . 2 (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
2 inss1 4158 . . . . . . . . 9 (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
32unissi 4813 . . . . . . . 8 (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
43sseli 3913 . . . . . . 7 (𝑦 (𝐵 ∩ 𝒫 𝑥) → 𝑦 𝐵)
54pm4.71ri 564 . . . . . 6 (𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ (𝑦 𝐵𝑦 (𝐵 ∩ 𝒫 𝑥)))
65ralbii 3133 . . . . 5 (∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦𝑥 (𝑦 𝐵𝑦 (𝐵 ∩ 𝒫 𝑥)))
7 r19.26 3137 . . . . 5 (∀𝑦𝑥 (𝑦 𝐵𝑦 (𝐵 ∩ 𝒫 𝑥)) ↔ (∀𝑦𝑥 𝑦 𝐵 ∧ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥)))
86, 7bitri 278 . . . 4 (∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ (∀𝑦𝑥 𝑦 𝐵 ∧ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥)))
9 dfss3 3905 . . . 4 (𝑥 (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥))
10 dfss3 3905 . . . . 5 (𝑥 𝐵 ↔ ∀𝑦𝑥 𝑦 𝐵)
11 elin 3899 . . . . . . . . . . 11 (𝑧 ∈ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑧𝐵𝑧 ∈ 𝒫 𝑥))
1211anbi2i 625 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑦𝑧 ∧ (𝑧𝐵𝑧 ∈ 𝒫 𝑥)))
13 an12 644 . . . . . . . . . 10 ((𝑦𝑧 ∧ (𝑧𝐵𝑧 ∈ 𝒫 𝑥)) ↔ (𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
1412, 13bitri 278 . . . . . . . . 9 ((𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
1514exbii 1849 . . . . . . . 8 (∃𝑧(𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ ∃𝑧(𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
16 eluni 4807 . . . . . . . 8 (𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧(𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)))
17 df-rex 3112 . . . . . . . 8 (∃𝑧𝐵 (𝑦𝑧𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧(𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
1815, 16, 173bitr4i 306 . . . . . . 7 (𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧𝐵 (𝑦𝑧𝑧 ∈ 𝒫 𝑥))
19 velpw 4505 . . . . . . . . 9 (𝑧 ∈ 𝒫 𝑥𝑧𝑥)
2019anbi2i 625 . . . . . . . 8 ((𝑦𝑧𝑧 ∈ 𝒫 𝑥) ↔ (𝑦𝑧𝑧𝑥))
2120rexbii 3211 . . . . . . 7 (∃𝑧𝐵 (𝑦𝑧𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
2218, 21bitr2i 279 . . . . . 6 (∃𝑧𝐵 (𝑦𝑧𝑧𝑥) ↔ 𝑦 (𝐵 ∩ 𝒫 𝑥))
2322ralbii 3133 . . . . 5 (∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥) ↔ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥))
2410, 23anbi12i 629 . . . 4 ((𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)) ↔ (∀𝑦𝑥 𝑦 𝐵 ∧ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥)))
258, 9, 243bitr4i 306 . . 3 (𝑥 (𝐵 ∩ 𝒫 𝑥) ↔ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
2625abbii 2863 . 2 {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))}
271, 26eqtrdi 2849 1 (𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4500  ∪ cuni 4804  ‘cfv 6332  topGenctg 16723 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6291  df-fun 6334  df-fv 6340  df-topgen 16729 This theorem is referenced by:  eltg2  21604
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