Theorem inpw 49300

Description: Two ways of expressing a collection of subsets as seen in df-ntr 22985, unimax 4887, 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 3897	. 2 ⊢ (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝒫 𝐵}
2		elpw2g 5274	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵))
3	2	rabbidv 3396	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝒫 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})
4	1, 3	eqtrid 2783	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3389   ∩ cin 3888   ⊆ wss 3889  𝒫 cpw 4541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-in 3896  df-ss 3906  df-pw 4543

This theorem is referenced by:  toplatglb  49476