Theorem ovmpoa

Description: Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013)

	Ref	Expression
Hypotheses	ovmpoga.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑅 = 𝑆 )
	ovmpoga.2	⊢ 𝐹 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝑅 )
	ovmpoa.4	⊢ 𝑆 ∈ V
Assertion	ovmpoa	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( 𝐴 𝐹 𝐵 ) = 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		ovmpoga.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑅 = 𝑆 )
2		ovmpoga.2	⊢ 𝐹 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝑅 )
3		ovmpoa.4	⊢ 𝑆 ∈ V
4	1 2	ovmpoga	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V ) → ( 𝐴 𝐹 𝐵 ) = 𝑆 )
5	3 4	mp3an3	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( 𝐴 𝐹 𝐵 ) = 𝑆 )