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Theorem iinpw 3733
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 = x A 𝒫 x
Distinct variable group:   x,A

Proof of Theorem iinpw
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ssint 3622 . . . 4 (y Ax A yx)
2 vex 2554 . . . . . 6 y V
32elpw 3357 . . . . 5 (y 𝒫 xyx)
43ralbii 2324 . . . 4 (x A y 𝒫 xx A yx)
51, 4bitr4i 176 . . 3 (y Ax A y 𝒫 x)
62elpw 3357 . . 3 (y 𝒫 Ay A)
7 eliin 3653 . . . 4 (y V → (y x A 𝒫 xx A y 𝒫 x))
82, 7ax-mp 7 . . 3 (y x A 𝒫 xx A y 𝒫 x)
95, 6, 83bitr4i 201 . 2 (y 𝒫 Ay x A 𝒫 x)
109eqriv 2034 1 𝒫 A = x A 𝒫 x
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242   wcel 1390  wral 2300  Vcvv 2551  wss 2911  𝒫 cpw 3351   cint 3606   ciin 3649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-int 3607  df-iin 3651
This theorem is referenced by: (None)
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