Theorem rr-elrnmpt3d

Description: Elementhood in an image set. (Contributed by Rohan Ridenour, 11-Aug-2023)

	Ref	Expression
Hypotheses	rr-elrnmpt3d.1	|- F = ( x e. A |-> B )
	rr-elrnmpt3d.2	|- ( ph -> C e. A )
	rr-elrnmpt3d.3	|- ( ph -> D e. V )
	rr-elrnmpt3d.4	|- ( ( ph /\ x = C ) -> B = D )
Assertion	rr-elrnmpt3d	|- ( ph -> D e. ran F )

Proof

Step	Hyp	Ref	Expression
1		rr-elrnmpt3d.1	|- F = ( x e. A |-> B )
2		rr-elrnmpt3d.2	|- ( ph -> C e. A )
3		rr-elrnmpt3d.3	|- ( ph -> D e. V )
4		rr-elrnmpt3d.4	|- ( ( ph /\ x = C ) -> B = D )
5	4	eqcomd	|- ( ( ph /\ x = C ) -> D = B )
6	1 2 3 5	elrnmptdv	|- ( ph -> D e. ran F )