Theorem newbdayim

Description: One direction of the biconditional in newbday . (Contributed by Scott Fenton, 7-Nov-2025)

	Ref	Expression
Assertion	newbdayim	|- ( X e. ( _New ` A ) -> ( bday ` X ) = A )

Proof

Step	Hyp	Ref	Expression
1		elfvdm	|- ( X e. ( _New ` A ) -> A e. dom _New )
2		newf	|- _New : On --> ~P No
3		fdm	|- ( _New : On --> ~P No -> dom _New = On )
4	2 3	ax-mp	|- dom _New = On
5	1 4	eleqtrdi	|- ( X e. ( _New ` A ) -> A e. On )
6		newssno	|- ( _New ` A ) C_ No
7	6	sseli	|- ( X e. ( _New ` A ) -> X e. No )
8		newbday	|- ( ( A e. On /\ X e. No ) -> ( X e. ( _New ` A ) <-> ( bday ` X ) = A ) )
9	5 7 8	syl2anc	|- ( X e. ( _New ` A ) -> ( X e. ( _New ` A ) <-> ( bday ` X ) = A ) )
10	9	ibi	|- ( X e. ( _New ` A ) -> ( bday ` X ) = A )