Theorem iinpw 2617

Description: The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33.

Assertion

Ref	Expression
iinpw	|- P~|^|A = |^|_x e. A P~x

Distinct variable group:   x,A

Proof of Theorem iinpw

Step	Hyp	Ref	Expression
1		ssint 2549	. . . 4 |- (y (_ |^|A <-> A.x e. A y (_ x)
2		visset 1813	. . . . . 6 |- y e. V
3	2	elpw 2404	. . . . 5 |- (y e. P~x <-> y (_ x)
4	3	ralbii 1667	. . . 4 |- (A.x e. A y e. P~x <-> A.x e. A y (_ x)
5	1, 4	bitr4 176	. . 3 |- (y (_ |^|A <-> A.x e. A y e. P~x)
6	2	elpw 2404	. . 3 |- (y e. P~|^|A <-> y (_ |^|A)
7		eliin 2571	. . . 4 |- (y e. V -> (y e. |^|_x e. A P~x <-> A.x e. A y e. P~x))
8	2, 7	ax-mp 7	. . 3 |- (y e. |^|_x e. A P~x <-> A.x e. A y e. P~x)
9	5, 6, 8	3bitr4 183	. 2 |- (y e. P~|^|A <-> y e. |^|_x e. A P~x)
10	9	eqriv 1474	1 |- P~|^|A = |^|_x e. A P~x

Syntax hints:   <-> wb 146   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811   (_ wss 2047  P~cpw 2401  |^|cint 2533  |^|_ciin 2567

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-pw 2402  df-int 2534  df-iin 2569

Copyright terms: Public domain