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Metamath Proof Explorer

Theorem fmptd

Description: Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013)

Ref Expression
Hypotheses fmptd.1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝐶 )
fmptd.2 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
Assertion fmptd ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐶 )

Proof

Step Hyp Ref Expression
1 fmptd.1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝐶 )
2 fmptd.2 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
3 1 ralrimiva ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 )
4 2 fmpt ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹 : 𝐴 ⟶ 𝐶 )
5 3 4 sylib ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐶 )