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Theorem iinpw 3716
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 3605 . . . 4 (y Ax A yx)
2 vex 2538 . . . . . 6 y V
32elpw 3340 . . . . 5 (y 𝒫 xyx)
43ralbii 2308 . . . 4 (x A y 𝒫 xx A yx)
51, 4bitr4i 176 . . 3 (y Ax A y 𝒫 x)
62elpw 3340 . . 3 (y 𝒫 Ay A)
7 eliin 3636 . . . 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 2019 1 𝒫 A = x A 𝒫 x
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1228   wcel 1374  wral 2284  Vcvv 2535  wss 2894  𝒫 cpw 3334   cint 3589   ciin 3632
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336  df-int 3590  df-iin 3634
This theorem is referenced by: (None)
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