Theorem rnmptsn

Description: The range of a function mapping to singletons. (Contributed by ML, 15-Jul-2020)

	Ref	Expression
Assertion	rnmptsn	|- ran ( x e. A |-> { x } ) = { u | E. x e. A u = { x } }

Proof

Step	Hyp	Ref	Expression
1		rnopab	|- ran { <. x , u >. | ( x e. A /\ u = { x } ) } = { u | E. x ( x e. A /\ u = { x } ) }
2		df-mpt	|- ( x e. A |-> { x } ) = { <. x , u >. | ( x e. A /\ u = { x } ) }
3	2	rneqi	|- ran ( x e. A |-> { x } ) = ran { <. x , u >. | ( x e. A /\ u = { x } ) }
4		df-rex	|- ( E. x e. A u = { x } <-> E. x ( x e. A /\ u = { x } ) )
5	4	abbii	|- { u | E. x e. A u = { x } } = { u | E. x ( x e. A /\ u = { x } ) }
6	1 3 5	3eqtr4i	|- ran ( x e. A |-> { x } ) = { u | E. x e. A u = { x } }