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Metamath Proof Explorer

Theorem ovmpod

Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014)

Ref Expression
Hypotheses ovmpod.1 ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝑅 ) )
ovmpod.2 ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → 𝑅 = 𝑆 )
ovmpod.3 ⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
ovmpod.4 ⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
ovmpod.5 ⊢ ( 𝜑 → 𝑆 ∈ 𝑋 )
Assertion ovmpod ( 𝜑 → ( 𝐴 𝐹 𝐵 ) = 𝑆 )

Proof

Step Hyp Ref Expression
1 ovmpod.1 ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝑅 ) )
2 ovmpod.2 ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → 𝑅 = 𝑆 )
3 ovmpod.3 ⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
4 ovmpod.4 ⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
5 ovmpod.5 ⊢ ( 𝜑 → 𝑆 ∈ 𝑋 )
6 eqidd ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐷 = 𝐷 )
7 1 2 6 3 4 5 ovmpodx ⊢ ( 𝜑 → ( 𝐴 𝐹 𝐵 ) = 𝑆 )