Theorem newval

Description: The value of the new options function. (Contributed by Scott Fenton, 6-Aug-2024)

	Ref	Expression
Assertion	newval	$$⊢A\in \mathrm{On}\to N\left(A\right)=M\left(A\right)\setminus \mathrm{Old}\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		fveq2	$$⊢x=A\to M\left(x\right)=M\left(A\right)$$
2		fveq2	$$⊢x=A\to \mathrm{Old}\left(x\right)=\mathrm{Old}\left(A\right)$$
3	1 2	difeq12d	$$⊢x=A\to M\left(x\right)\setminus \mathrm{Old}\left(x\right)=M\left(A\right)\setminus \mathrm{Old}\left(A\right)$$
4		df-new	$$⊢N=\left(x\in \mathrm{On}⟼M\left(x\right)\setminus \mathrm{Old}\left(x\right)\right)$$
5		fvex	$$⊢M\left(A\right)\in \mathrm{V}$$
6	5	difexi	$$⊢M\left(A\right)\setminus \mathrm{Old}\left(A\right)\in \mathrm{V}$$
7	3 4 6	fvmpt	$$⊢A\in \mathrm{On}\to N\left(A\right)=M\left(A\right)\setminus \mathrm{Old}\left(A\right)$$