Theorem bj-iminvval

Description: Value of the functionalized inverse image. (Contributed by BJ, 23-May-2024)

	Ref	Expression
Hypotheses	bj-iminvval.1	$$⊢\phi \to A\in U$$
	bj-iminvval.2	$$⊢\phi \to B\in V$$
Assertion	bj-iminvval	$$⊢\phi \to A{𝒫}^{*}B=\left(r\in 𝒫\left(A\times B\right)⟼\left\{⟨x,y⟩|\left(\left(x\subseteq A\wedge y\subseteq B\right)\wedge x=r^{-1}\left[y\right]\right)\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-iminvval.1	$$⊢\phi \to A\in U$$
2		bj-iminvval.2	$$⊢\phi \to B\in V$$
3		df-iminv	$$⊢{𝒫}^{*}=\left(a\in \mathrm{V},b\in \mathrm{V}⟼\left(r\in 𝒫\left(a\times b\right)⟼\left\{⟨x,y⟩|\left(\left(x\subseteq a\wedge y\subseteq b\right)\wedge x=r^{-1}\left[y\right]\right)\right\}\right)\right)$$
4	1 2 3	bj-imdirvallem	$$⊢\phi \to A{𝒫}^{*}B=\left(r\in 𝒫\left(A\times B\right)⟼\left\{⟨x,y⟩|\left(\left(x\subseteq A\wedge y\subseteq B\right)\wedge x=r^{-1}\left[y\right]\right)\right\}\right)$$