Theorem rrx2pyel

Description: The y-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023)

	Ref	Expression
Hypotheses	rrx2px.i	|- I = { 1 , 2 }
	rrx2px.b	|- P = ( RR ^m I )
Assertion	rrx2pyel	|- ( X e. P -> ( X ` 2 ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		rrx2px.i	|- I = { 1 , 2 }
2		rrx2px.b	|- P = ( RR ^m I )
3		id	|- ( X e. P -> X e. P )
4		2ex	|- 2 e. _V
5	4	prid2	|- 2 e. { 1 , 2 }
6	5 1	eleqtrri	|- 2 e. I
7	6	a1i	|- ( X e. P -> 2 e. I )
8	2 3 7	mapfvd	|- ( X e. P -> ( X ` 2 ) e. RR )