Theorem elrnmpti

Description: Membership in the range of a function. (Contributed by NM, 30-Aug-2004) (Revised by Mario Carneiro, 31-Aug-2015)

	Ref	Expression
Hypotheses	rnmpt.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
	elrnmpti.2	⊢ 𝐵 ∈ V
Assertion	elrnmpti	⊢ ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		rnmpt.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
2		elrnmpti.2	⊢ 𝐵 ∈ V
3	2	rgenw	⊢ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ V
4	1	elrnmptg	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ V → ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 ) )
5	3 4	ax-mp	⊢ ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 )