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Metamath Proof Explorer

Theorem inpw

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

		Ref	Expression
	Assertion	inpw	$⊢ B \in V \to A \cap 𝒫 B = \{x \in A \| x \subseteq B\}$

Step	Hyp	Ref	Expression
1		dfin5	$⊢ A \cap 𝒫 B = \{x \in A \| x \in 𝒫 B\}$
2		elpw2g	$⊢ B \in V \to (x \in 𝒫 B \leftrightarrow x \subseteq B)$
3	2	rabbidv	$⊢ B \in V \to \{x \in A \| x \in 𝒫 B\} = \{x \in A \| x \subseteq B\}$
4	1 3	eqtrid	$⊢ B \in V \to A \cap 𝒫 B = \{x \in A \| x \subseteq B\}$