Theorem ffvelrn

Description: A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999)

	Ref	Expression
Assertion	ffvelrn	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐹 ‘ 𝐶 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ffn	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 Fn 𝐴 )
2		fnfvelrn	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐹 ‘ 𝐶 ) ∈ ran 𝐹 )
3	1 2	sylan	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐹 ‘ 𝐶 ) ∈ ran 𝐹 )
4		frn	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ran 𝐹 ⊆ 𝐵 )
5	4	sseld	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( ( 𝐹 ‘ 𝐶 ) ∈ ran 𝐹 → ( 𝐹 ‘ 𝐶 ) ∈ 𝐵 ) )
6	5	adantr	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝐶 ) ∈ ran 𝐹 → ( 𝐹 ‘ 𝐶 ) ∈ 𝐵 ) )
7	3 6	mpd	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐹 ‘ 𝐶 ) ∈ 𝐵 )