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Metamath Proof Explorer

Theorem fnmptd

Description: The maps-to notation defines a function with domain. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses fnmptd.1 ⊢ Ⅎ 𝑥 𝜑
fnmptd.2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
fnmptd.3 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
Assertion fnmptd ( 𝜑 → 𝐹 Fn 𝐴 )

Proof

Step Hyp Ref Expression
1 fnmptd.1 ⊢ Ⅎ 𝑥 𝜑
2 fnmptd.2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
3 fnmptd.3 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
4 2 ex ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉 ) )
5 1 4 ralrimi ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 )
6 3 fnmpt ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴 )
7 5 6 syl ⊢ ( 𝜑 → 𝐹 Fn 𝐴 )