Theorem 1stinl

Description: The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022)

	Ref	Expression
Assertion	1stinl	$$⊢X\in V\to 1^{st}\left(inl\left(X\right)\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		df-inl	$$⊢inl=\left(x\in \mathrm{V}⟼⟨\varnothing ,x⟩\right)$$
2		opeq2	$$⊢x=X\to ⟨\varnothing ,x⟩=⟨\varnothing ,X⟩$$
3		elex	$$⊢X\in V\to X\in \mathrm{V}$$
4		opex	$$⊢⟨\varnothing ,X⟩\in \mathrm{V}$$
5	4	a1i	$$⊢X\in V\to ⟨\varnothing ,X⟩\in \mathrm{V}$$
6	1 2 3 5	fvmptd3	$$⊢X\in V\to inl\left(X\right)=⟨\varnothing ,X⟩$$
7	6	fveq2d	$$⊢X\in V\to 1^{st}\left(inl\left(X\right)\right)=1^{st}\left(⟨\varnothing ,X⟩\right)$$
8		0ex	$$⊢\varnothing \in \mathrm{V}$$
9		op1stg	$$⊢\left(\varnothing \in \mathrm{V}\wedge X\in V\right)\to 1^{st}\left(⟨\varnothing ,X⟩\right)=\varnothing$$
10	8 9	mpan	$$⊢X\in V\to 1^{st}\left(⟨\varnothing ,X⟩\right)=\varnothing$$
11	7 10	eqtrd	$$⊢X\in V\to 1^{st}\left(inl\left(X\right)\right)=\varnothing$$