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	$$⊢I=\left\{1,2\right\}$$
	rrx2px.b	$$⊢P={ℝ}^I$$
Assertion	rrx2pxel	$$⊢X\in P\to X\left(1\right)\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		rrx2px.i	$$⊢I=\left\{1,2\right\}$$
2		rrx2px.b	$$⊢P={ℝ}^I$$
3		id	$$⊢X\in P\to X\in P$$
4		1ex	$$⊢1\in \mathrm{V}$$
5	4	prid1	$$⊢1\in \left\{1,2\right\}$$
6	5 1	eleqtrri	$$⊢1\in I$$
7	6	a1i	$$⊢X\in P\to 1\in I$$
8	2 3 7	mapfvd	$$⊢X\in P\to X\left(1\right)\in ℝ$$