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Theorem iinpw 4769
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 4645 . . . 4 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
2 selpw 4309 . . . . 5 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
32ralbii 3118 . . . 4 (∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∀𝑥𝐴 𝑦𝑥)
41, 3bitr4i 267 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
5 selpw 4309 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
6 vex 3343 . . . 4 𝑦 ∈ V
7 eliin 4677 . . . 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 2757 1 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  wss 3715  𝒫 cpw 4302   cint 4627   ciin 4673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342  df-in 3722  df-ss 3729  df-pw 4304  df-int 4628  df-iin 4675
This theorem is referenced by: (None)
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