Theorem fvelimabd

Description: Deduction form of fvelimab . (Contributed by Stanislas Polu, 9-Mar-2020)

	Ref	Expression
Hypotheses	fvelimabd.1	⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
	fvelimabd.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
Assertion	fvelimabd	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐹 “ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		fvelimabd.1	⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
2		fvelimabd.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
3		fvelimab	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( 𝐶 ∈ ( 𝐹 “ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) = 𝐶 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐹 “ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) = 𝐶 ) )