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	$$⊢I=\left(0..^2\right)$$
	rrx2px.b	$$⊢P={ℝ}^I$$
Assertion	rrx2pyel	$$⊢X\in P\to X\left(1\right)\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		rrx2px.i	$$⊢I=\left(0..^2\right)$$
2		rrx2px.b	$$⊢P={ℝ}^I$$
3		id	$$⊢X\in P\to X\in P$$
4		1elpr01	$$⊢1\in \left\{0,1\right\}$$
5		fzo0to2pr	$$⊢\left(0..^2\right)=\left\{0,1\right\}$$
6	1 5	eqtri	$$⊢I=\left\{0,1\right\}$$
7	4 6	eleqtrri	$$⊢1\in I$$
8	7	a1i	$$⊢X\in P\to 1\in I$$
9	2 3 8	mapfvd	$$⊢X\in P\to X\left(1\right)\in ℝ$$