Theorem newf

Description: The new function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024)

	Ref	Expression
Assertion	newf	$$⊢N:\mathrm{On}⟶𝒫\mathrm{No}$$

Proof

Step	Hyp	Ref	Expression
1		df-new	$$⊢N=\left(x\in \mathrm{On}⟼M\left(x\right)\setminus \mathrm{Old}\left(x\right)\right)$$
2		madef	$$⊢M:\mathrm{On}⟶𝒫\mathrm{No}$$
3	2	ffvelcdmi	$$⊢x\in \mathrm{On}\to M\left(x\right)\in 𝒫\mathrm{No}$$
4	3	elpwid	$$⊢x\in \mathrm{On}\to M\left(x\right)\subseteq \mathrm{No}$$
5	4	ssdifssd	$$⊢x\in \mathrm{On}\to M\left(x\right)\setminus \mathrm{Old}\left(x\right)\subseteq \mathrm{No}$$
6		fvex	$$⊢M\left(x\right)\in \mathrm{V}$$
7	6	difexi	$$⊢M\left(x\right)\setminus \mathrm{Old}\left(x\right)\in \mathrm{V}$$
8	7	elpw	$$⊢M\left(x\right)\setminus \mathrm{Old}\left(x\right)\in 𝒫\mathrm{No}\leftrightarrow M\left(x\right)\setminus \mathrm{Old}\left(x\right)\subseteq \mathrm{No}$$
9	5 8	sylibr	$$⊢x\in \mathrm{On}\to M\left(x\right)\setminus \mathrm{Old}\left(x\right)\in 𝒫\mathrm{No}$$
10	1 9	fmpti	$$⊢N:\mathrm{On}⟶𝒫\mathrm{No}$$