Theorem elpmrn

Description: The range of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Assertion	elpmrn	⊢ ( 𝐹 ∈ ( 𝐴 ↑pm 𝐵 ) → ran 𝐹 ⊆ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		elpmi	⊢ ( 𝐹 ∈ ( 𝐴 ↑pm 𝐵 ) → ( 𝐹 : dom 𝐹 ⟶ 𝐴 ∧ dom 𝐹 ⊆ 𝐵 ) )
2	1	simpld	⊢ ( 𝐹 ∈ ( 𝐴 ↑pm 𝐵 ) → 𝐹 : dom 𝐹 ⟶ 𝐴 )
3	2	frnd	⊢ ( 𝐹 ∈ ( 𝐴 ↑pm 𝐵 ) → ran 𝐹 ⊆ 𝐴 )