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Theorem tgval2 20954
Description: Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 20967) that (topGen‘𝐵) is indeed a topology (on 𝐵, see unitg 20965). See also tgval 20953 and tgval3 20961. (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 20953 . 2 (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
2 inss1 3968 . . . . . . . . 9 (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
32unissi 4605 . . . . . . . 8 (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
43sseli 3732 . . . . . . 7 (𝑦 (𝐵 ∩ 𝒫 𝑥) → 𝑦 𝐵)
54pm4.71ri 668 . . . . . 6 (𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ (𝑦 𝐵𝑦 (𝐵 ∩ 𝒫 𝑥)))
65ralbii 3110 . . . . 5 (∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦𝑥 (𝑦 𝐵𝑦 (𝐵 ∩ 𝒫 𝑥)))
7 r19.26 3194 . . . . 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 3725 . . . 4 (𝑥 (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥))
10 dfss3 3725 . . . . 5 (𝑥 𝐵 ↔ ∀𝑦𝑥 𝑦 𝐵)
11 elin 3931 . . . . . . . . . . 11 (𝑧 ∈ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑧𝐵𝑧 ∈ 𝒫 𝑥))
1211anbi2i 732 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑦𝑧 ∧ (𝑧𝐵𝑧 ∈ 𝒫 𝑥)))
13 an12 873 . . . . . . . . . 10 ((𝑦𝑧 ∧ (𝑧𝐵𝑧 ∈ 𝒫 𝑥)) ↔ (𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
1412, 13bitri 264 . . . . . . . . 9 ((𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
1514exbii 1915 . . . . . . . 8 (∃𝑧(𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ ∃𝑧(𝑧𝐵 ∧ (𝑦𝑧𝑧 ∈ 𝒫 𝑥)))
16 eluni 4583 . . . . . . . 8 (𝑦 (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧(𝑦𝑧𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)))
17 df-rex 3048 . . . . . . . 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 4301 . . . . . . . . 9 (𝑧 ∈ 𝒫 𝑥𝑧𝑥)
2019anbi2i 732 . . . . . . . 8 ((𝑦𝑧𝑧 ∈ 𝒫 𝑥) ↔ (𝑦𝑧𝑧𝑥))
2120rexbii 3171 . . . . . . 7 (∃𝑧𝐵 (𝑦𝑧𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧𝐵 (𝑦𝑧𝑧𝑥))
2218, 21bitr2i 265 . . . . . 6 (∃𝑧𝐵 (𝑦𝑧𝑧𝑥) ↔ 𝑦 (𝐵 ∩ 𝒫 𝑥))
2322ralbii 3110 . . . . 5 (∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥) ↔ ∀𝑦𝑥 𝑦 (𝐵 ∩ 𝒫 𝑥))
2410, 23anbi12i 735 . . . 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 2869 . 2 {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))}
271, 26syl6eq 2802 1 (𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wex 1845  wcel 2131  {cab 2738  wral 3042  wrex 3043  cin 3706  wss 3707  𝒫 cpw 4294   cuni 4580  cfv 6041  topGenctg 16292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049  df-topgen 16298
This theorem is referenced by:  eltg2  20956
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