Theorem inpw 50044

Description: Two ways of expressing a collection of subsets as seen in df-ntr 23347, unimax 4904, 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 4091	. 2 ⊢ (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝒫 𝐵}
2		elpw2g 5295	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵))
3	2	rabbidv 3419	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝒫 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})
4	1, 3	eqtrid 2807	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {crab 3412   ⊆ wss 3896   ∩ cin 4088  𝒫 cpw 4556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-in 4090  df-pw 4558

This theorem is used by:  toplatglb  50223