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 = ( 0 ..^ 2 )
	rrx2px.b	⊢ 𝑃 = ( ℝ ↑m 𝐼 )
Assertion	rrx2pyel	⊢ ( 𝑋 ∈ 𝑃 → ( 𝑋 ‘ 1 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		rrx2px.i	⊢ 𝐼 = ( 0 ..^ 2 )
2		rrx2px.b	⊢ 𝑃 = ( ℝ ↑m 𝐼 )
3		id	⊢ ( 𝑋 ∈ 𝑃 → 𝑋 ∈ 𝑃 )
4		1elpr01	⊢ 1 ∈ { 0 , 1 }
5		fzo0to2pr	⊢ ( 0 ..^ 2 ) = { 0 , 1 }
6	1 5	eqtri	⊢ 𝐼 = { 0 , 1 }
7	4 6	eleqtrri	⊢ 1 ∈ 𝐼
8	7	a1i	⊢ ( 𝑋 ∈ 𝑃 → 1 ∈ 𝐼 )
9	2 3 8	mapfvd	⊢ ( 𝑋 ∈ 𝑃 → ( 𝑋 ‘ 1 ) ∈ ℝ )