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Theorem iinpw 4751
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 4627 . . . 4 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
2 selpw 4304 . . . . 5 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
32ralbii 3129 . . . 4 (∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∀𝑥𝐴 𝑦𝑥)
41, 3bitr4i 267 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
5 selpw 4304 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
6 vex 3354 . . . 4 𝑦 ∈ V
7 eliin 4659 . . . 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 292 . 2 (𝑦 ∈ 𝒫 𝐴𝑦 𝑥𝐴 𝒫 𝑥)
109eqriv 2768 1 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  wss 3723  𝒫 cpw 4297   cint 4611   ciin 4655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-in 3730  df-ss 3737  df-pw 4299  df-int 4612  df-iin 4657
This theorem is referenced by: (None)
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