Theorem lrrecval2

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

	Ref	Expression
Hypothesis	lrrec.1	|- R = { <. x , y >. | x e. ( ( _L ` y ) u. ( _R ` y ) ) }
Assertion	lrrecval2	|- ( ( A e. No /\ B e. No ) -> ( A R B <-> ( bday ` A ) e. ( bday ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		lrrec.1	|- R = { <. x , y >. | x e. ( ( _L ` y ) u. ( _R ` y ) ) }
2	1	lrrecval	|- ( ( A e. No /\ B e. No ) -> ( A R B <-> A e. ( ( _L ` B ) u. ( _R ` B ) ) ) )
3		lrold	|- ( B e. No -> ( ( _L ` B ) u. ( _R ` B ) ) = ( _Old ` ( bday ` B ) ) )
4	3	adantl	|- ( ( A e. No /\ B e. No ) -> ( ( _L ` B ) u. ( _R ` B ) ) = ( _Old ` ( bday ` B ) ) )
5	4	eleq2d	|- ( ( A e. No /\ B e. No ) -> ( A e. ( ( _L ` B ) u. ( _R ` B ) ) <-> A e. ( _Old ` ( bday ` B ) ) ) )
6		bdayelon	|- ( bday ` B ) e. On
7		oldbday	|- ( ( ( bday ` B ) e. On /\ A e. No ) -> ( A e. ( _Old ` ( bday ` B ) ) <-> ( bday ` A ) e. ( bday ` B ) ) )
8	6 7	mpan	|- ( A e. No -> ( A e. ( _Old ` ( bday ` B ) ) <-> ( bday ` A ) e. ( bday ` B ) ) )
9	8	adantr	|- ( ( A e. No /\ B e. No ) -> ( A e. ( _Old ` ( bday ` B ) ) <-> ( bday ` A ) e. ( bday ` B ) ) )
10	2 5 9	3bitrd	|- ( ( A e. No /\ B e. No ) -> ( A R B <-> ( bday ` A ) e. ( bday ` B ) ) )