Theorem iinpw 5071

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 4929	. . . 4 ⊢ (𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
2		velpw 4565	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑥 ↔ 𝑦 ⊆ 𝑥)
3	2	ralbii 3112	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
4	1, 3	bitr4i 281	. . 3 ⊢ (𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)
5		velpw 4565	. . 3 ⊢ (𝑦 ∈ 𝒫 ∩ 𝐴 ↔ 𝑦 ⊆ ∩ 𝐴)
6		eliin 4961	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥))
7	6	elv 3459	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)
8	4, 5, 7	3bitr4i 306	. 2 ⊢ (𝑦 ∈ 𝒫 ∩ 𝐴 ↔ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥)
9	8	eqriv 2760	1 ⊢ 𝒫 ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝒫 𝑥

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560  ∩ cint 4912  ∩ ciin 4957

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-v 3456  df-ss 3901  df-pw 4562  df-int 4913  df-iin 4959

This theorem is used by: (None)