Theorem fvixp2

Description: Projection of a factor of an indexed Cartesian product. (Contributed by Glauco Siliprandi, 24-Dec-2020)

	Ref	Expression
Assertion	fvixp2	⊢ ( ( 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elixp2	⊢ ( 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 ↔ ( 𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) )
2	1	simp3bi	⊢ ( 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 )
3	2	r19.21bi	⊢ ( ( 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 )