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 ) )