Theorem rightval

Description: The value of the right options function. (Contributed by Scott Fenton, 9-Oct-2024)

	Ref	Expression
Assertion	rightval	$$⊢R\left(A\right)=\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|A{<}_{s}x\right\}$$

Proof

Step	Hyp	Ref	Expression
1		2fveq3	$$⊢y=A\to \mathrm{Old}\left(\mathrm{bday}\left(y\right)\right)=\mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)$$
2		breq1	$$⊢y=A\to \left(y{<}_{s}x\leftrightarrow A{<}_{s}x\right)$$
3	1 2	rabeqbidv	$$⊢y=A\to \left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(y\right)\right)|y{<}_{s}x\right\}=\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|A{<}_{s}x\right\}$$
4		df-right	$$⊢R=\left(y\in \mathrm{No}⟼\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(y\right)\right)|y{<}_{s}x\right\}\right)$$
5		fvex	$$⊢\mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)\in \mathrm{V}$$
6	5	rabex	$$⊢\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|A{<}_{s}x\right\}\in \mathrm{V}$$
7	3 4 6	fvmpt	$$⊢A\in \mathrm{No}\to R\left(A\right)=\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|A{<}_{s}x\right\}$$
8	4	fvmptndm	$$⊢\neg A\in \mathrm{No}\to R\left(A\right)=\varnothing$$
9		bdaydm	$$⊢\mathrm{dom}\mathrm{bday}=\mathrm{No}$$
10	9	eleq2i	$$⊢A\in \mathrm{dom}\mathrm{bday}\leftrightarrow A\in \mathrm{No}$$
11		ndmfv	$$⊢\neg A\in \mathrm{dom}\mathrm{bday}\to \mathrm{bday}\left(A\right)=\varnothing$$
12	10 11	sylnbir	$$⊢\neg A\in \mathrm{No}\to \mathrm{bday}\left(A\right)=\varnothing$$
13	12	fveq2d	$$⊢\neg A\in \mathrm{No}\to \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)=\mathrm{Old}\left(\varnothing \right)$$
14		old0	$$⊢\mathrm{Old}\left(\varnothing \right)=\varnothing$$
15	13 14	eqtrdi	$$⊢\neg A\in \mathrm{No}\to \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)=\varnothing$$
16	15	rabeqdv	$$⊢\neg A\in \mathrm{No}\to \left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|A{<}_{s}x\right\}=\left\{x\in \varnothing |A{<}_{s}x\right\}$$
17		rab0	$$⊢\left\{x\in \varnothing |A{<}_{s}x\right\}=\varnothing$$
18	16 17	eqtrdi	$$⊢\neg A\in \mathrm{No}\to \left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|A{<}_{s}x\right\}=\varnothing$$
19	8 18	eqtr4d	$$⊢\neg A\in \mathrm{No}\to R\left(A\right)=\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|A{<}_{s}x\right\}$$
20	7 19	pm2.61i	$$⊢R\left(A\right)=\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|A{<}_{s}x\right\}$$