Theorem fvmptelrn

Description: The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypothesis	fvmptelrn.1	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ 𝐶 )
Assertion	fvmptelrn	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		fvmptelrn.1	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ 𝐶 )
2		eqid	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
3	2	fmpt	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ 𝐶 )
4	1 3	sylibr	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 )
5	4	r19.21bi	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝐶 )