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 ∈ 𝑉 → ( 1st ‘ ( inl ‘ 𝑋 ) ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		df-inl	⊢ inl = ( 𝑥 ∈ V ↦ ⟨ ∅ , 𝑥 ⟩ )
2		opeq2	⊢ ( 𝑥 = 𝑋 → ⟨ ∅ , 𝑥 ⟩ = ⟨ ∅ , 𝑋 ⟩ )
3		elex	⊢ ( 𝑋 ∈ 𝑉 → 𝑋 ∈ V )
4		opex	⊢ ⟨ ∅ , 𝑋 ⟩ ∈ V
5	4	a1i	⊢ ( 𝑋 ∈ 𝑉 → ⟨ ∅ , 𝑋 ⟩ ∈ V )
6	1 2 3 5	fvmptd3	⊢ ( 𝑋 ∈ 𝑉 → ( inl ‘ 𝑋 ) = ⟨ ∅ , 𝑋 ⟩ )
7	6	fveq2d	⊢ ( 𝑋 ∈ 𝑉 → ( 1st ‘ ( inl ‘ 𝑋 ) ) = ( 1st ‘ ⟨ ∅ , 𝑋 ⟩ ) )
8		0ex	⊢ ∅ ∈ V
9		op1stg	⊢ ( ( ∅ ∈ V ∧ 𝑋 ∈ 𝑉 ) → ( 1st ‘ ⟨ ∅ , 𝑋 ⟩ ) = ∅ )
10	8 9	mpan	⊢ ( 𝑋 ∈ 𝑉 → ( 1st ‘ ⟨ ∅ , 𝑋 ⟩ ) = ∅ )
11	7 10	eqtrd	⊢ ( 𝑋 ∈ 𝑉 → ( 1st ‘ ( inl ‘ 𝑋 ) ) = ∅ )