Theorem iinpw 5019

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 4883	. . . 4 ⊢ (𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
2		velpw 4545	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑥 ↔ 𝑦 ⊆ 𝑥)
3	2	ralbii 3163	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
4	1, 3	bitr4i 280	. . 3 ⊢ (𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)
5		velpw 4545	. . 3 ⊢ (𝑦 ∈ 𝒫 ∩ 𝐴 ↔ 𝑦 ⊆ ∩ 𝐴)
6		eliin 4915	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥))
7	6	elv 3498	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)
8	4, 5, 7	3bitr4i 305	. 2 ⊢ (𝑦 ∈ 𝒫 ∩ 𝐴 ↔ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥)
9	8	eqriv 2816	1 ⊢ 𝒫 ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝒫 𝑥

Syntax hints:   ↔ wb 208   = wceq 1530   ∈ wcel 2107  ∀wral 3136  Vcvv 3493   ⊆ wss 3934  𝒫 cpw 4537  ∩ cint 4867  ∩ ciin 4911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-v 3495  df-in 3941  df-ss 3950  df-pw 4539  df-int 4868  df-iin 4913

This theorem is referenced by: (None)