Theorem fvmptrabfv

Description: Value of a function mapping a set to a class abstraction restricting the value of another function. (Contributed by AV, 18-Feb-2022)

	Ref	Expression
Hypotheses	fvmptrabfv.f	$$⊢F=\left(x\in \mathrm{V}⟼\left\{y\in G\left(x\right)|\phi \right\}\right)$$
	fvmptrabfv.r	$$⊢x=X\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	fvmptrabfv	$$⊢F\left(X\right)=\left\{y\in G\left(X\right)|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		fvmptrabfv.f	$$⊢F=\left(x\in \mathrm{V}⟼\left\{y\in G\left(x\right)|\phi \right\}\right)$$
2		fvmptrabfv.r	$$⊢x=X\to \left(\phi \leftrightarrow \psi \right)$$
3		fveq2	$$⊢x=X\to G\left(x\right)=G\left(X\right)$$
4	3 2	rabeqbidv	$$⊢x=X\to \left\{y\in G\left(x\right)|\phi \right\}=\left\{y\in G\left(X\right)|\psi \right\}$$
5		fvex	$$⊢G\left(X\right)\in \mathrm{V}$$
6	5	rabex	$$⊢\left\{y\in G\left(X\right)|\psi \right\}\in \mathrm{V}$$
7	4 1 6	fvmpt	$$⊢X\in \mathrm{V}\to F\left(X\right)=\left\{y\in G\left(X\right)|\psi \right\}$$
8		fvprc	$$⊢\neg X\in \mathrm{V}\to F\left(X\right)=\varnothing$$
9		fvprc	$$⊢\neg X\in \mathrm{V}\to G\left(X\right)=\varnothing$$
10	9	rabeqdv	$$⊢\neg X\in \mathrm{V}\to \left\{y\in G\left(X\right)|\psi \right\}=\left\{y\in \varnothing |\psi \right\}$$
11		rab0	$$⊢\left\{y\in \varnothing |\psi \right\}=\varnothing$$
12	10 11	eqtr2di	$$⊢\neg X\in \mathrm{V}\to \varnothing =\left\{y\in G\left(X\right)|\psi \right\}$$
13	8 12	eqtrd	$$⊢\neg X\in \mathrm{V}\to F\left(X\right)=\left\{y\in G\left(X\right)|\psi \right\}$$
14	7 13	pm2.61i	$$⊢F\left(X\right)=\left\{y\in G\left(X\right)|\psi \right\}$$