Theorem ovmpog

Description: Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 14-Sep-1999) (Revised by David Abernethy, 19-Jun-2012)

	Ref	Expression
Hypotheses	ovmpog.1	⊢ ( 𝑥 = 𝐴 → 𝑅 = 𝐺 )
	ovmpog.2	⊢ ( 𝑦 = 𝐵 → 𝐺 = 𝑆 )
	ovmpog.3	⊢ 𝐹 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝑅 )
Assertion	ovmpog	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻 ) → ( 𝐴 𝐹 𝐵 ) = 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		ovmpog.1	⊢ ( 𝑥 = 𝐴 → 𝑅 = 𝐺 )
2		ovmpog.2	⊢ ( 𝑦 = 𝐵 → 𝐺 = 𝑆 )
3		ovmpog.3	⊢ 𝐹 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝑅 )
4	1 2	sylan9eq	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑅 = 𝑆 )
5	4 3	ovmpoga	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻 ) → ( 𝐴 𝐹 𝐵 ) = 𝑆 )