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Theorem tgval2 11919
 Description: Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 11932) that (topGen‘𝐵) is indeed a topology (on ∪ 𝐵, see unitg 11930). See also tgval 11917 and tgval3 11926. (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 11917 . 2 (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
2 inss1 3235 . . . . . . . . 9 (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
32unissi 3698 . . . . . . . 8 (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
43sseli 3035 . . . . . . 7 (𝑦 (𝐵 ∩ 𝒫 𝑥) → 𝑦 𝐵)
54pm4.71ri 385 . . . . . 6 (𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ (𝑦 𝐵𝑦 (𝐵 ∩ 𝒫 𝑥)))
65ralbii 2395 . . . . 5 (∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦𝑥 (𝑦 𝐵𝑦 (𝐵 ∩ 𝒫 𝑥)))
7 r19.26 2511 . . . . 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 3029 . . . 4 (𝑥 (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥))
10 dfss3 3029 . . . . 5 (𝑥 𝐵 ↔ ∀𝑦𝑥 𝑦 𝐵)
11 elin 3198 . . . . . . . . . . 11 (𝑧 ∈ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑧𝐵𝑧 ∈ 𝒫 𝑥))
1211anbi2i 446 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑦𝑧 ∧ (𝑧𝐵𝑧 ∈ 𝒫 𝑥)))
13 an12 529 . . . . . . . . . 10 ((𝑦𝑧 ∧ (𝑧𝐵𝑧 ∈ 𝒫 𝑥)) ↔ (𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
1412, 13bitri 183 . . . . . . . . 9 ((𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
1514exbii 1548 . . . . . . . 8 (∃𝑧(𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ ∃𝑧(𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
16 eluni 3678 . . . . . . . 8 (𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧(𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)))
17 df-rex 2376 . . . . . . . 8 (∃𝑧𝐵 (𝑦𝑧𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧(𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
1815, 16, 173bitr4i 211 . . . . . . 7 (𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧𝐵 (𝑦𝑧𝑧 ∈ 𝒫 𝑥))
19 selpw 3456 . . . . . . . . 9 (𝑧 ∈ 𝒫 𝑥𝑧𝑥)
2019anbi2i 446 . . . . . . . 8 ((𝑦𝑧𝑧 ∈ 𝒫 𝑥) ↔ (𝑦𝑧𝑧𝑥))
2120rexbii 2396 . . . . . . 7 (∃𝑧𝐵 (𝑦𝑧𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
2218, 21bitr2i 184 . . . . . 6 (∃𝑧𝐵 (𝑦𝑧𝑧𝑥) ↔ 𝑦 (𝐵 ∩ 𝒫 𝑥))
2322ralbii 2395 . . . . 5 (∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥) ↔ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥))
2410, 23anbi12i 449 . . . 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 2210 . 2 {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))}
271, 26syl6eq 2143 1 (𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1296  ∃wex 1433   ∈ wcel 1445  {cab 2081  ∀wral 2370  ∃wrex 2371   ∩ cin 3012   ⊆ wss 3013  𝒫 cpw 3449  ∪ cuni 3675  ‘cfv 5049  topGenctg 11835 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-topgen 11841 This theorem is referenced by:  eltg2  11921
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