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Theorem tgval2 12234
Description: Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 12247) that (topGen‘𝐵) is indeed a topology (on 𝐵, see unitg 12245). See also tgval 12232 and tgval3 12241. (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 12232 . 2 (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
2 inss1 3296 . . . . . . . . 9 (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
32unissi 3759 . . . . . . . 8 (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
43sseli 3093 . . . . . . 7 (𝑦 (𝐵 ∩ 𝒫 𝑥) → 𝑦 𝐵)
54pm4.71ri 389 . . . . . 6 (𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ (𝑦 𝐵𝑦 (𝐵 ∩ 𝒫 𝑥)))
65ralbii 2441 . . . . 5 (∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦𝑥 (𝑦 𝐵𝑦 (𝐵 ∩ 𝒫 𝑥)))
7 r19.26 2558 . . . . 5 (∀𝑦𝑥 (𝑦 𝐵𝑦 (𝐵 ∩ 𝒫 𝑥)) ↔ (∀𝑦𝑥 𝑦 𝐵 ∧ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥)))
86, 7bitri 183 . . . 4 (∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ (∀𝑦𝑥 𝑦 𝐵 ∧ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥)))
9 dfss3 3087 . . . 4 (𝑥 (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥))
10 dfss3 3087 . . . . 5 (𝑥 𝐵 ↔ ∀𝑦𝑥 𝑦 𝐵)
11 elin 3259 . . . . . . . . . . 11 (𝑧 ∈ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑧𝐵𝑧 ∈ 𝒫 𝑥))
1211anbi2i 452 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑦𝑧 ∧ (𝑧𝐵𝑧 ∈ 𝒫 𝑥)))
13 an12 550 . . . . . . . . . 10 ((𝑦𝑧 ∧ (𝑧𝐵𝑧 ∈ 𝒫 𝑥)) ↔ (𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
1412, 13bitri 183 . . . . . . . . 9 ((𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
1514exbii 1584 . . . . . . . 8 (∃𝑧(𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ ∃𝑧(𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
16 eluni 3739 . . . . . . . 8 (𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧(𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)))
17 df-rex 2422 . . . . . . . 8 (∃𝑧𝐵 (𝑦𝑧𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧(𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
1815, 16, 173bitr4i 211 . . . . . . 7 (𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧𝐵 (𝑦𝑧𝑧 ∈ 𝒫 𝑥))
19 velpw 3517 . . . . . . . . 9 (𝑧 ∈ 𝒫 𝑥𝑧𝑥)
2019anbi2i 452 . . . . . . . 8 ((𝑦𝑧𝑧 ∈ 𝒫 𝑥) ↔ (𝑦𝑧𝑧𝑥))
2120rexbii 2442 . . . . . . 7 (∃𝑧𝐵 (𝑦𝑧𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
2218, 21bitr2i 184 . . . . . 6 (∃𝑧𝐵 (𝑦𝑧𝑧𝑥) ↔ 𝑦 (𝐵 ∩ 𝒫 𝑥))
2322ralbii 2441 . . . . 5 (∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥) ↔ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥))
2410, 23anbi12i 455 . . . 4 ((𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)) ↔ (∀𝑦𝑥 𝑦 𝐵 ∧ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥)))
258, 9, 243bitr4i 211 . . 3 (𝑥 (𝐵 ∩ 𝒫 𝑥) ↔ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
2625abbii 2255 . 2 {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))}
271, 26syl6eq 2188 1 (𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wex 1468  wcel 1480  {cab 2125  wral 2416  wrex 2417  cin 3070  wss 3071  𝒫 cpw 3510   cuni 3736  cfv 5123  topGenctg 12149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-topgen 12155
This theorem is referenced by:  eltg2  12236
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