Theorem iinpw 3972

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 3856	. . . 4 ⊢ (𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
2		vex 2738	. . . . . 6 ⊢ 𝑦 ∈ V
3	2	elpw 3578	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑥 ↔ 𝑦 ⊆ 𝑥)
4	3	ralbii 2481	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
5	1, 4	bitr4i 187	. . 3 ⊢ (𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)
6	2	elpw 3578	. . 3 ⊢ (𝑦 ∈ 𝒫 ∩ 𝐴 ↔ 𝑦 ⊆ ∩ 𝐴)
7		eliin 3887	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥))
8	2, 7	ax-mp 5	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)
9	5, 6, 8	3bitr4i 212	. 2 ⊢ (𝑦 ∈ 𝒫 ∩ 𝐴 ↔ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥)
10	9	eqriv 2172	1 ⊢ 𝒫 ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝒫 𝑥

Syntax hints:   ↔ wb 105   = wceq 1353   ∈ wcel 2146  ∀wral 2453  Vcvv 2735   ⊆ wss 3127  𝒫 cpw 3572  ∩ cint 3840  ∩ ciin 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-v 2737  df-in 3133  df-ss 3140  df-pw 3574  df-int 3841  df-iin 3885

This theorem is referenced by: (None)