Theorem leftval

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

	Ref	Expression
Assertion	leftval	$$⊢A\in \mathrm{No}\to L\left(A\right)=\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|x{<}_{s}A\right\}$$

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		breq2	$$⊢y=A\to \left(x{<}_{s}y\leftrightarrow x{<}_{s}A\right)$$
3	1 2	rabeqbidv	$$⊢y=A\to \left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(y\right)\right)|x{<}_{s}y\right\}=\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|x{<}_{s}A\right\}$$
4		df-left	$$⊢L=\left(y\in \mathrm{No}⟼\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(y\right)\right)|x{<}_{s}y\right\}\right)$$
5		fvex	$$⊢\mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)\in \mathrm{V}$$
6	5	rabex	$$⊢\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|x{<}_{s}A\right\}\in \mathrm{V}$$
7	3 4 6	fvmpt	$$⊢A\in \mathrm{No}\to L\left(A\right)=\left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|x{<}_{s}A\right\}$$