Theorem iinpw 4054

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	⊢ ℘∩A = ∩x ∈ A ℘x

Distinct variable group:   x,A

Proof of Theorem iinpw

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssint 3942	. . . 4 ⊢ (y ⊆ ∩A ↔ ∀x ∈ A y ⊆ x)
2		vex 2862	. . . . . 6 ⊢ y ∈ V
3	2	elpw 3728	. . . . 5 ⊢ (y ∈ ℘x ↔ y ⊆ x)
4	3	ralbii 2638	. . . 4 ⊢ (∀x ∈ A y ∈ ℘x ↔ ∀x ∈ A y ⊆ x)
5	1, 4	bitr4i 243	. . 3 ⊢ (y ⊆ ∩A ↔ ∀x ∈ A y ∈ ℘x)
6	2	elpw 3728	. . 3 ⊢ (y ∈ ℘∩A ↔ y ⊆ ∩A)
7		eliin 3974	. . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ A ℘x ↔ ∀x ∈ A y ∈ ℘x))
8	2, 7	ax-mp 5	. . 3 ⊢ (y ∈ ∩x ∈ A ℘x ↔ ∀x ∈ A y ∈ ℘x)
9	5, 6, 8	3bitr4i 268	. 2 ⊢ (y ∈ ℘∩A ↔ y ∈ ∩x ∈ A ℘x)
10	9	eqriv 2350	1 ⊢ ℘∩A = ∩x ∈ A ℘x

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2614  Vcvv 2859   ⊆ wss 3257  ℘cpw 3722  ∩cint 3926  ∩ciin 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pw 3724  df-int 3927  df-iin 3972

This theorem is referenced by: (None)