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 = { 1 , 2 }
	rrx2px.b	⊢ 𝑃 = ( ℝ ↑m 𝐼 )
Assertion	rrx2pxel	⊢ ( 𝑋 ∈ 𝑃 → ( 𝑋 ‘ 1 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		rrx2px.i	⊢ 𝐼 = { 1 , 2 }
2		rrx2px.b	⊢ 𝑃 = ( ℝ ↑m 𝐼 )
3		id	⊢ ( 𝑋 ∈ 𝑃 → 𝑋 ∈ 𝑃 )
4		1ex	⊢ 1 ∈ V
5	4	prid1	⊢ 1 ∈ { 1 , 2 }
6	5 1	eleqtrri	⊢ 1 ∈ 𝐼
7	6	a1i	⊢ ( 𝑋 ∈ 𝑃 → 1 ∈ 𝐼 )
8	2 3 7	mapfvd	⊢ ( 𝑋 ∈ 𝑃 → ( 𝑋 ‘ 1 ) ∈ ℝ )