Theorem 1stinr

Description: The first component of the value of a right injection is 1o . (Contributed by AV, 27-Jun-2022)

	Ref	Expression
Assertion	1stinr	$$⊢X\in V\to 1^{st}\left(inr\left(X\right)\right)=1_{𝑜}$$

Proof

Step	Hyp	Ref	Expression
1		df-inr	$$⊢inr=\left(x\in \mathrm{V}⟼⟨1_{𝑜},x⟩\right)$$
2		opeq2	$$⊢x=X\to ⟨1_{𝑜},x⟩=⟨1_{𝑜},X⟩$$
3		elex	$$⊢X\in V\to X\in \mathrm{V}$$
4		opex	$$⊢⟨1_{𝑜},X⟩\in \mathrm{V}$$
5	4	a1i	$$⊢X\in V\to ⟨1_{𝑜},X⟩\in \mathrm{V}$$
6	1 2 3 5	fvmptd3	$$⊢X\in V\to inr\left(X\right)=⟨1_{𝑜},X⟩$$
7	6	fveq2d	$$⊢X\in V\to 1^{st}\left(inr\left(X\right)\right)=1^{st}\left(⟨1_{𝑜},X⟩\right)$$
8		1oex	$$⊢1_{𝑜}\in \mathrm{V}$$
9		op1stg	$$⊢\left(1_{𝑜}\in \mathrm{V}\wedge X\in V\right)\to 1^{st}\left(⟨1_{𝑜},X⟩\right)=1_{𝑜}$$
10	8 9	mpan	$$⊢X\in V\to 1^{st}\left(⟨1_{𝑜},X⟩\right)=1_{𝑜}$$
11	7 10	eqtrd	$$⊢X\in V\to 1^{st}\left(inr\left(X\right)\right)=1_{𝑜}$$