Theorem leftval

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

	Ref	Expression
Assertion	leftval	Could not format assertion : No typesetting found for |- ( _Left ` A ) = { x e. ( _Old ` ( bday ` A ) ) | x

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	Could not format _Left = ( y e. No |-> { x e. ( _Old ` ( bday ` y ) ) | x { x e. ( _Old ` ( bday ` y ) ) | x
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	Could not format ( A e. No -> ( _Left ` A ) = { x e. ( _Old ` ( bday ` A ) ) | x ( _Left ` A ) = { x e. ( _Old ` ( bday ` A ) ) | x
8	4	fvmptndm	Could not format ( -. A e. No -> ( _Left ` A ) = (/) ) : No typesetting found for |- ( -. A e. No -> ( _Left ` A ) = (/) ) with typecode |-
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	$$⊢\neg A\in \mathrm{No}\to \left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|x{<}_{s}A\right\}=\left\{x\in \varnothing |x{<}_{s}A\right\}$$
17		rab0	$$⊢\left\{x\in \varnothing |x{<}_{s}A\right\}=\varnothing$$
18	16 17	eqtrdi	$$⊢\neg A\in \mathrm{No}\to \left\{x\in \mathrm{Old}\left(\mathrm{bday}\left(A\right)\right)|x{<}_{s}A\right\}=\varnothing$$
19	8 18	eqtr4d	Could not format ( -. A e. No -> ( _Left ` A ) = { x e. ( _Old ` ( bday ` A ) ) | x ( _Left ` A ) = { x e. ( _Old ` ( bday ` A ) ) | x
20	7 19	pm2.61i	Could not format ( _Left ` A ) = { x e. ( _Old ` ( bday ` A ) ) | x