Theorem lrrecval2

Description: Next, we establish an alternate expression for R . (Contributed by Scott Fenton, 19-Aug-2024)

	Ref	Expression
Hypothesis	lrrec.1	No typesetting found for |- R = { <. x , y >. | x e. ( ( _Left ` y ) u. ( _Right ` y ) ) } with typecode |-
Assertion	lrrecval2	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(ARB\leftrightarrow \mathrm{bday}\left(A\right)\in \mathrm{bday}\left(B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		lrrec.1	Could not format R = { <. x , y >. | x e. ( ( _Left ` y ) u. ( _Right ` y ) ) } : No typesetting found for |- R = { <. x , y >. | x e. ( ( _Left ` y ) u. ( _Right ` y ) ) } with typecode |-
2	1	lrrecval	Could not format ( ( A e. No /\ B e. No ) -> ( A R B <-> A e. ( ( _Left ` B ) u. ( _Right ` B ) ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( A R B <-> A e. ( ( _Left ` B ) u. ( _Right ` B ) ) ) ) with typecode |-
3		lrold	Could not format ( ( _Left ` B ) u. ( _Right ` B ) ) = ( _Old ` ( bday ` B ) ) : No typesetting found for |- ( ( _Left ` B ) u. ( _Right ` B ) ) = ( _Old ` ( bday ` B ) ) with typecode |-
4	3	a1i	Could not format ( ( A e. No /\ B e. No ) -> ( ( _Left ` B ) u. ( _Right ` B ) ) = ( _Old ` ( bday ` B ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( ( _Left ` B ) u. ( _Right ` B ) ) = ( _Old ` ( bday ` B ) ) ) with typecode |-
5	4	eleq2d	Could not format ( ( A e. No /\ B e. No ) -> ( A e. ( ( _Left ` B ) u. ( _Right ` B ) ) <-> A e. ( _Old ` ( bday ` B ) ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( A e. ( ( _Left ` B ) u. ( _Right ` B ) ) <-> A e. ( _Old ` ( bday ` B ) ) ) ) with typecode |-
6		bdayelon	$$⊢\mathrm{bday}\left(B\right)\in \mathrm{On}$$
7		oldbday	$$⊢\left(\mathrm{bday}\left(B\right)\in \mathrm{On}\wedge A\in \mathrm{No}\right)\to \left(A\in \mathrm{Old}\left(\mathrm{bday}\left(B\right)\right)\leftrightarrow \mathrm{bday}\left(A\right)\in \mathrm{bday}\left(B\right)\right)$$
8	6 7	mpan	$$⊢A\in \mathrm{No}\to \left(A\in \mathrm{Old}\left(\mathrm{bday}\left(B\right)\right)\leftrightarrow \mathrm{bday}\left(A\right)\in \mathrm{bday}\left(B\right)\right)$$
9	8	adantr	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A\in \mathrm{Old}\left(\mathrm{bday}\left(B\right)\right)\leftrightarrow \mathrm{bday}\left(A\right)\in \mathrm{bday}\left(B\right)\right)$$
10	2 5 9	3bitrd	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(ARB\leftrightarrow \mathrm{bday}\left(A\right)\in \mathrm{bday}\left(B\right)\right)$$