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Theorem iinpw 4550
Description: The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
Assertion
Ref Expression
iinpw 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥
Distinct variable group:   𝑥,𝐴

Proof of Theorem iinpw
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssint 4428 . . . 4 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
2 selpw 4115 . . . . 5 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
32ralbii 2963 . . . 4 (∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∀𝑥𝐴 𝑦𝑥)
41, 3bitr4i 266 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
5 selpw 4115 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
6 vex 3176 . . . 4 𝑦 ∈ V
7 eliin 4461 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥))
86, 7ax-mp 5 . . 3 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
94, 5, 83bitr4i 291 . 2 (𝑦 ∈ 𝒫 𝐴𝑦 𝑥𝐴 𝒫 𝑥)
109eqriv 2607 1 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥
Colors of variables: wff setvar class
Syntax hints:  wb 195   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  wss 3540  𝒫 cpw 4108   cint 4410   ciin 4456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-int 4411  df-iin 4458
This theorem is referenced by: (None)
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