Theorem iinpw 2038

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(s):   x,A

Proof of Theorem iinpw

Step	Hyp	Ref	Expression
1		ssint 1980	. . . 4 |- (y (_ |^|A <-> A.x e. A y (_ x)
2		visset 1350	. . . . . 6 |- y e. V
3	2	elpw 1801	. . . . 5 |- (y e. P~x <-> y (_ x)
4	3	biral 1223	. . . 4 |- (A.x e. A y e. P~x <-> A.x e. A y (_ x)
5	1, 4	bitr4 154	. . 3 |- (y (_ |^|A <-> A.x e. A y e. P~x)
6	2	elpw 1801	. . 3 |- (y e. P~|^|A <-> y (_ |^|A)
7		eliin 1999	. . . 4 |- (y e. V -> (y e. |^|x e. A P~x <-> A.x e. A y e. P~x))
8	2, 7	ax-mp 6	. . 3 |- (y e. |^|x e. A P~x <-> A.x e. A y e. P~x)
9	5, 6, 8	3bitr4 158	. 2 |- (y e. P~|^|A <-> y e. |^|x e. A P~x)
10	9	cleqri 1101	1 |- P~|^|A = |^|x e. A P~x

Syntax hints:   <-> wb 127   = wceq 1091   e. wcel 1092  A.wral 1201  Vcvv 1348   (_ wss 1487  P~cpw 1798  |^|cint 1965  |^|ciin 1995

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-pw 1799  df-int 1966  df-iin 1997

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