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 )