Theorem iinpw 5031

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.

Step	Hyp	Ref	Expression
1		ssint 4892	. . . 4 ⊢ (𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
2		velpw 4535	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑥 ↔ 𝑦 ⊆ 𝑥)
3	2	ralbii 3090	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
4	1, 3	bitr4i 277	. . 3 ⊢ (𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)
5		velpw 4535	. . 3 ⊢ (𝑦 ∈ 𝒫 ∩ 𝐴 ↔ 𝑦 ⊆ ∩ 𝐴)
6		eliin 4926	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥))
7	6	elv 3428	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)
8	4, 5, 7	3bitr4i 302	. 2 ⊢ (𝑦 ∈ 𝒫 ∩ 𝐴 ↔ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥)
9	8	eqriv 2735	1 ⊢ 𝒫 ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝒫 𝑥

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  ∩ cint 4876  ∩ ciin 4922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-v 3424  df-in 3890  df-ss 3900  df-pw 4532  df-int 4877  df-iin 4924

This theorem is referenced by: (None)