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Metamath Proof Explorer

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 ) )