Theorem inpw 48738

Description: Two ways of expressing a collection of subsets as seen in df-ntr 23028, unimax 4944, and others (Contributed by Zhi Wang, 27-Sep-2024.)

Assertion

Ref	Expression
inpw	⊢ (𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉

Proof of Theorem inpw

Step	Hyp	Ref	Expression
1		dfin5 3959	. 2 ⊢ (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝒫 𝐵}
2		elpw2g 5333	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵))
3	2	rabbidv 3444	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝒫 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})
4	1, 3	eqtrid 2789	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  {crab 3436   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-pw 4602

This theorem is referenced by:  toplatglb  48890