Theorem inpw 49481

Description: Two ways of expressing a collection of subsets as seen in df-ntr 23142, unimax 4911, 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 3921	. 2 ⊢ (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝒫 𝐵}
2		elpw2g 5301	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵))
3	2	rabbidv 3430	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝒫 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})
4	1, 3	eqtrid 2816	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  {crab 3423   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-in 3920  df-ss 3930  df-pw 4566

This theorem is referenced by:  toplatglb  49657