Theorem elrnmpog

Description: Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007) (Revised by Mario Carneiro, 31-Aug-2015)

	Ref	Expression
Hypothesis	rngop.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
Assertion	elrnmpog	⊢ ( 𝐷 ∈ 𝑉 → ( 𝐷 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐷 = 𝐶 ) )

Proof

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