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Metamath Proof Explorer

Theorem ovmpoga

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

Ref Expression
Hypotheses ovmpoga.1 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑅 = 𝑆 )
ovmpoga.2 ⊢ 𝐹 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝑅 )
Assertion ovmpoga ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻 ) → ( 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 elex ⊢ ( 𝑆 ∈ 𝐻 → 𝑆 ∈ V )
4 2 a1i ⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V ) → 𝐹 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝑅 ) )
5 1 adantl ⊢ ( ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V ) ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → 𝑅 = 𝑆 )
6 simp1 ⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V ) → 𝐴 ∈ 𝐶 )
7 simp2 ⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V ) → 𝐵 ∈ 𝐷 )
8 simp3 ⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V ) → 𝑆 ∈ V )
9 4 5 6 7 8 ovmpod ⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V ) → ( 𝐴 𝐹 𝐵 ) = 𝑆 )
10 3 9 syl3an3 ⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻 ) → ( 𝐴 𝐹 𝐵 ) = 𝑆 )