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Theorem iinpw 3992
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 3875 . . . 4 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
2 vex 2755 . . . . . 6 𝑦 ∈ V
32elpw 3596 . . . . 5 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
43ralbii 2496 . . . 4 (∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∀𝑥𝐴 𝑦𝑥)
51, 4bitr4i 187 . . 3 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
62elpw 3596 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
7 eliin 3906 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥))
82, 7ax-mp 5 . . 3 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∀𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
95, 6, 83bitr4i 212 . 2 (𝑦 ∈ 𝒫 𝐴𝑦 𝑥𝐴 𝒫 𝑥)
109eqriv 2186 1 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1364  wcel 2160  wral 2468  Vcvv 2752  wss 3144  𝒫 cpw 3590   cint 3859   ciin 3902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-int 3860  df-iin 3904
This theorem is referenced by: (None)
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