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Metamath Proof Explorer

Theorem ovig

Description: The value of an operation class abstraction (weak version). (Contributed by NM, 14-Sep-1999) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012) (Revised by Mario Carneiro, 19-Dec-2013)

Ref Expression
Hypotheses ovig.1 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝜑 ↔ 𝜓 ) )
ovig.2 ⊢ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) → ∃* 𝑧 𝜑 )
ovig.3 ⊢ 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) }
Assertion ovig ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷 ) → ( 𝜓 → ( 𝐴 𝐹 𝐵 ) = 𝐶 ) )

Proof

Step Hyp Ref Expression
1 ovig.1 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝜑 ↔ 𝜓 ) )
2 ovig.2 ⊢ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) → ∃* 𝑧 𝜑 )
3 ovig.3 ⊢ 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) }
4 3simpa ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷 ) → ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) )
5 eleq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝑅 ↔ 𝐴 ∈ 𝑅 ) )
6 eleq1 ⊢ ( 𝑦 = 𝐵 → ( 𝑦 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆 ) )
7 5 6 bi2anan9 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) )
8 7 3adant3 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) )
9 8 1 anbi12d ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜓 ) ) )
10 moanimv ⊢ ( ∃* 𝑧 ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) ↔ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) → ∃* 𝑧 𝜑 ) )
11 2 10 mpbir ⊢ ∃* 𝑧 ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 )
12 9 11 3 ovigg ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷 ) → ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜓 ) → ( 𝐴 𝐹 𝐵 ) = 𝐶 ) )
13 4 12 mpand ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷 ) → ( 𝜓 → ( 𝐴 𝐹 𝐵 ) = 𝐶 ) )