Theorem ovmpox2

Description: The value of an operation class abstraction. Variant of ovmpoga which does not require D and x to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010) (Revised by Mario Carneiro, 20-Dec-2013)

	Ref	Expression
Hypotheses	ovmpox2.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑅 = 𝑆 )
	ovmpox2.2	⊢ ( 𝑦 = 𝐵 → 𝐶 = 𝐿 )
	ovmpox2.3	⊢ 𝐹 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝑅 )
Assertion	ovmpox2	⊢ ( ( 𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻 ) → ( 𝐴 𝐹 𝐵 ) = 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		ovmpox2.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑅 = 𝑆 )
2		ovmpox2.2	⊢ ( 𝑦 = 𝐵 → 𝐶 = 𝐿 )
3		ovmpox2.3	⊢ 𝐹 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝑅 )
4	3	a1i	⊢ ( ( 𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻 ) → 𝐹 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐷 ↦ 𝑅 ) )
5	1	adantl	⊢ ( ( ( 𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻 ) ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → 𝑅 = 𝑆 )
6	2	adantl	⊢ ( ( ( 𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻 ) ∧ 𝑦 = 𝐵 ) → 𝐶 = 𝐿 )
7		simp1	⊢ ( ( 𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻 ) → 𝐴 ∈ 𝐿 )
8		simp2	⊢ ( ( 𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻 ) → 𝐵 ∈ 𝐷 )
9		simp3	⊢ ( ( 𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻 ) → 𝑆 ∈ 𝐻 )
10	4 5 6 7 8 9	ovmpordx	⊢ ( ( 𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻 ) → ( 𝐴 𝐹 𝐵 ) = 𝑆 )