Theorem elrnmpt

Description: The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015)

	Ref	Expression
Hypothesis	rnmpt.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
Assertion	elrnmpt	⊢ ( 𝐶 ∈ 𝑉 → ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		rnmpt.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
2		eqeq1	⊢ ( 𝑦 = 𝐶 → ( 𝑦 = 𝐵 ↔ 𝐶 = 𝐵 ) )
3	2	rexbidv	⊢ ( 𝑦 = 𝐶 → ( ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 ) )
4	1	rnmpt	⊢ ran 𝐹 = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 }
5	3 4	elab2g	⊢ ( 𝐶 ∈ 𝑉 → ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 ) )