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Theorem tgval2 20808
Description: Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 20821) that (topGen‘𝐵) is indeed a topology (on 𝐵, see unitg 20819). See also tgval 20807 and tgval3 20815. (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 20807 . 2 (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
2 inss1 3866 . . . . . . . . 9 (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
32unissi 4493 . . . . . . . 8 (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
43sseli 3632 . . . . . . 7 (𝑦 (𝐵 ∩ 𝒫 𝑥) → 𝑦 𝐵)
54pm4.71ri 666 . . . . . 6 (𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ (𝑦 𝐵𝑦 (𝐵 ∩ 𝒫 𝑥)))
65ralbii 3009 . . . . 5 (∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦𝑥 (𝑦 𝐵𝑦 (𝐵 ∩ 𝒫 𝑥)))
7 r19.26 3093 . . . . 5 (∀𝑦𝑥 (𝑦 𝐵𝑦 (𝐵 ∩ 𝒫 𝑥)) ↔ (∀𝑦𝑥 𝑦 𝐵 ∧ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥)))
86, 7bitri 264 . . . 4 (∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ (∀𝑦𝑥 𝑦 𝐵 ∧ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥)))
9 dfss3 3625 . . . 4 (𝑥 (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥))
10 dfss3 3625 . . . . 5 (𝑥 𝐵 ↔ ∀𝑦𝑥 𝑦 𝐵)
11 elin 3829 . . . . . . . . . . 11 (𝑧 ∈ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑧𝐵𝑧 ∈ 𝒫 𝑥))
1211anbi2i 730 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑦𝑧 ∧ (𝑧𝐵𝑧 ∈ 𝒫 𝑥)))
13 an12 855 . . . . . . . . . 10 ((𝑦𝑧 ∧ (𝑧𝐵𝑧 ∈ 𝒫 𝑥)) ↔ (𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
1412, 13bitri 264 . . . . . . . . 9 ((𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
1514exbii 1814 . . . . . . . 8 (∃𝑧(𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ ∃𝑧(𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
16 eluni 4471 . . . . . . . 8 (𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧(𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)))
17 df-rex 2947 . . . . . . . 8 (∃𝑧𝐵 (𝑦𝑧𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧(𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
1815, 16, 173bitr4i 292 . . . . . . 7 (𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧𝐵 (𝑦𝑧𝑧 ∈ 𝒫 𝑥))
19 selpw 4198 . . . . . . . . 9 (𝑧 ∈ 𝒫 𝑥𝑧𝑥)
2019anbi2i 730 . . . . . . . 8 ((𝑦𝑧𝑧 ∈ 𝒫 𝑥) ↔ (𝑦𝑧𝑧𝑥))
2120rexbii 3070 . . . . . . 7 (∃𝑧𝐵 (𝑦𝑧𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
2218, 21bitr2i 265 . . . . . 6 (∃𝑧𝐵 (𝑦𝑧𝑧𝑥) ↔ 𝑦 (𝐵 ∩ 𝒫 𝑥))
2322ralbii 3009 . . . . 5 (∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥) ↔ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥))
2410, 23anbi12i 733 . . . 4 ((𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)) ↔ (∀𝑦𝑥 𝑦 𝐵 ∧ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥)))
258, 9, 243bitr4i 292 . . 3 (𝑥 (𝐵 ∩ 𝒫 𝑥) ↔ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
2625abbii 2768 . 2 {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))}
271, 26syl6eq 2701 1 (𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  wrex 2942  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468  cfv 5926  topGenctg 16145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-topgen 16151
This theorem is referenced by:  eltg2  20810
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