Theorem rrx2pxel

Description: The x-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 = ( 0 ..^ 2 )
	rrx2px.b	|- P = ( RR ^m I )
Assertion	rrx2pxel	|- ( X e. P -> ( X ` 0 ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		rrx2px.i	|- I = ( 0 ..^ 2 )
2		rrx2px.b	|- P = ( RR ^m I )
3		id	|- ( X e. P -> X e. P )
4		0elpr01	|- 0 e. { 0 , 1 }
5		fzo0to2pr	|- ( 0 ..^ 2 ) = { 0 , 1 }
6	1 5	eqtri	|- I = { 0 , 1 }
7	4 6	eleqtrri	|- 0 e. I
8	7	a1i	|- ( X e. P -> 0 e. I )
9	2 3 8	mapfvd	|- ( X e. P -> ( X ` 0 ) e. RR )