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Theorem tgval 21551
Description: The topology generated by a basis. See also tgval2 21552 and tgval3 21559. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
Assertion
Ref Expression
tgval (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
Distinct variable groups:   𝑥,𝐵   𝑥,𝑉

Proof of Theorem tgval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-topgen 16708 . 2 topGen = (𝑦 ∈ V ↦ {𝑥𝑥 (𝑦 ∩ 𝒫 𝑥)})
2 ineq1 4164 . . . . 5 (𝑦 = 𝐵 → (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
32unieqd 4835 . . . 4 (𝑦 = 𝐵 (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
43sseq2d 3983 . . 3 (𝑦 = 𝐵 → (𝑥 (𝑦 ∩ 𝒫 𝑥) ↔ 𝑥 (𝐵 ∩ 𝒫 𝑥)))
54abbidv 2888 . 2 (𝑦 = 𝐵 → {𝑥𝑥 (𝑦 ∩ 𝒫 𝑥)} = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
6 elex 3497 . 2 (𝐵𝑉𝐵 ∈ V)
7 uniexg 7451 . . 3 (𝐵𝑉 𝐵 ∈ V)
8 abssexg 5266 . . 3 ( 𝐵 ∈ V → {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V)
9 uniin 4845 . . . . . . 7 (𝐵 ∩ 𝒫 𝑥) ⊆ ( 𝐵 𝒫 𝑥)
10 sstr 3959 . . . . . . 7 ((𝑥 (𝐵 ∩ 𝒫 𝑥) ∧ (𝐵 ∩ 𝒫 𝑥) ⊆ ( 𝐵 𝒫 𝑥)) → 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
119, 10mpan2 690 . . . . . 6 (𝑥 (𝐵 ∩ 𝒫 𝑥) → 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
12 ssin 4190 . . . . . 6 ((𝑥 𝐵𝑥 𝒫 𝑥) ↔ 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
1311, 12sylibr 237 . . . . 5 (𝑥 (𝐵 ∩ 𝒫 𝑥) → (𝑥 𝐵𝑥 𝒫 𝑥))
1413ss2abi 4027 . . . 4 {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)}
15 ssexg 5210 . . . 4 (({𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∧ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V) → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
1614, 15mpan 689 . . 3 ({𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
177, 8, 163syl 18 . 2 (𝐵𝑉 → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
181, 5, 6, 17fvmptd3 6774 1 (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {cab 2802  Vcvv 3479  cin 3917  wss 3918  𝒫 cpw 4520   cuni 4821  cfv 6338  topGenctg 16702
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-iota 6297  df-fun 6340  df-fv 6346  df-topgen 16708
This theorem is referenced by:  tgval2  21552  eltg  21553  tgdif0  21588
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