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Theorem iinpw 3769
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 3658 . . . 4 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
2 vex 2577 . . . . . 6 𝑦 ∈ V
32elpw 3392 . . . . 5 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
43ralbii 2347 . . . 4 (∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∀𝑥𝐴 𝑦𝑥)
51, 4bitr4i 180 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
62elpw 3392 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
7 eliin 3689 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥))
82, 7ax-mp 7 . . 3 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
95, 6, 83bitr4i 205 . 2 (𝑦 ∈ 𝒫 𝐴𝑦 𝑥𝐴 𝒫 𝑥)
109eqriv 2053 1 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1259  wcel 1409  wral 2323  Vcvv 2574  wss 2944  𝒫 cpw 3386   cint 3642   ciin 3685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-in 2951  df-ss 2958  df-pw 3388  df-int 3643  df-iin 3687
This theorem is referenced by: (None)
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