Theorem catpropd

Description: Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017)

	Ref	Expression
Hypotheses	catpropd.1	⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
	catpropd.2	⊢ ( 𝜑 → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )
	catpropd.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
	catpropd.4	⊢ ( 𝜑 → 𝐷 ∈ 𝑊 )
Assertion	catpropd	⊢ ( 𝜑 → ( 𝐶 ∈ Cat ↔ 𝐷 ∈ Cat ) )

Proof

Step	Hyp	Ref	Expression
1		catpropd.1	⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
2		catpropd.2	⊢ ( 𝜑 → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )
3		catpropd.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
4		catpropd.4	⊢ ( 𝜑 → 𝐷 ∈ 𝑊 )
5		simpl	⊢ ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
6	5	2ralimi	⊢ ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
7	6	2ralimi	⊢ ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
8	7	adantl	⊢ ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
9	8	ralimi	⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
10	9	a1i	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
11		simpl	⊢ ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
12	11	2ralimi	⊢ ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
13	12	2ralimi	⊢ ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
14	13	adantl	⊢ ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
15	14	ralimi	⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
16	15	a1i	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
17		nfra1	⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 )
18		nfv	⊢ Ⅎ 𝑥 ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 )
19		nfra1	⊢ Ⅎ 𝑧 ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 )
20		nfv	⊢ Ⅎ 𝑦 ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 )
21		nfra1	⊢ Ⅎ 𝑔 ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 )
22		nfv	⊢ Ⅎ 𝑓 ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 )
23		oveq1	⊢ ( 𝑔 = ℎ → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) )
24	23	eleq1d	⊢ ( 𝑔 = ℎ → ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
25	24	cbvralvw	⊢ ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
26		oveq2	⊢ ( 𝑓 = 𝑔 → ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) )
27	26	eleq1d	⊢ ( 𝑓 = 𝑔 → ( ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
28	27	ralbidv	⊢ ( 𝑓 = 𝑔 → ( ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
29	25 28	syl5bb	⊢ ( 𝑓 = 𝑔 → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
30	21 22 29	cbvralw	⊢ ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
31		oveq2	⊢ ( 𝑧 = 𝑤 → ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) )
32		oveq2	⊢ ( 𝑧 = 𝑤 → ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) = ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) )
33	32	oveqd	⊢ ( 𝑧 = 𝑤 → ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) = ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) )
34		oveq2	⊢ ( 𝑧 = 𝑤 → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) )
35	33 34	eleq12d	⊢ ( 𝑧 = 𝑤 → ( ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
36	31 35	raleqbidv	⊢ ( 𝑧 = 𝑤 → ( ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
37	36	ralbidv	⊢ ( 𝑧 = 𝑤 → ( ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
38	30 37	syl5bb	⊢ ( 𝑧 = 𝑤 → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
39	38	cbvralvw	⊢ ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) )
40		oveq2	⊢ ( 𝑦 = 𝑧 → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
41		oveq1	⊢ ( 𝑦 = 𝑧 → ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) = ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) )
42		opeq2	⊢ ( 𝑦 = 𝑧 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑥 , 𝑧 ⟩ )
43	42	oveq1d	⊢ ( 𝑦 = 𝑧 → ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) = ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) )
44	43	oveqd	⊢ ( 𝑦 = 𝑧 → ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) )
45	44	eleq1d	⊢ ( 𝑦 = 𝑧 → ( ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
46	41 45	raleqbidv	⊢ ( 𝑦 = 𝑧 → ( ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
47	40 46	raleqbidv	⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
48	47	ralbidv	⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
49	39 48	syl5bb	⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
50	19 20 49	cbvralw	⊢ ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) )
51		oveq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) )
52		opeq1	⊢ ( 𝑥 = 𝑦 → ⟨ 𝑥 , 𝑧 ⟩ = ⟨ 𝑦 , 𝑧 ⟩ )
53	52	oveq1d	⊢ ( 𝑥 = 𝑦 → ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) = ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) )
54	53	oveqd	⊢ ( 𝑥 = 𝑦 → ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) = ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) )
55		oveq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) = ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) )
56	54 55	eleq12d	⊢ ( 𝑥 = 𝑦 → ( ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
57	56	ralbidv	⊢ ( 𝑥 = 𝑦 → ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
58	51 57	raleqbidv	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
59	58	ralbidv	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
60		ralcom	⊢ ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) )
61	59 60	bitrdi	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
62	61	ralbidv	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
63	50 62	syl5bb	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
64	17 18 63	cbvralw	⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) )
65	64	biimpi	⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) )
66	65	ancri	⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
67		r19.26	⊢ ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ↔ ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
68		r19.26	⊢ ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ↔ ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
69		r19.26	⊢ ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ↔ ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
70		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
71		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
72		eqid	⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 )
73		eqid	⊢ ( comp ‘ 𝐷 ) = ( comp ‘ 𝐷 )
74	1	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
75	74	ad4antr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
76	75	ad4antr	⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
77	2	ad5antr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )
78	77	ad4antr	⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )
79		simpllr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
80	79	ad2antrr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
81	80	ad4antr	⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
82		simp-4r	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) )
83	82	ad4antr	⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) )
84		simpllr	⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑤 ∈ ( Base ‘ 𝐶 ) )
85		simplr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) )
86	85	ad4antr	⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) )
87		simpr	⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) )
88	70 71 72 73 76 78 81 83 84 86 87	comfeqval	⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) )
89		simpllr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → 𝑧 ∈ ( Base ‘ 𝐶 ) )
90	89	ad4antr	⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑧 ∈ ( Base ‘ 𝐶 ) )
91		simp-4r	⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
92		simplr	⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) )
93	70 71 72 73 76 78 81 90 84 91 92	comfeqval	⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) )
94	88 93	eqeq12d	⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) )
95	94	ex	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) → ( ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
96	95	ralimdva	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) → ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
97		ralbi	⊢ ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) )
98	96 97	syl6	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) → ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
99	98	ralimdva	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) → ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
100	99	impancom	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
101	100	impr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) )
102		ralbi	⊢ ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) )
103	101 102	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) )
104	103	anbi2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
105	104	ex	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
106	105	ralimdva	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
107	69 106	syl5bir	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
108	107	expdimp	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
109		ralbi	⊢ ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
110	108 109	syl6	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
111	110	an32s	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
112	111	ralimdva	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
113		ralbi	⊢ ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
114	112 113	syl6	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
115	114	expimpd	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
116	115	ralimdva	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
117		ralbi	⊢ ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
118	116 117	syl6	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
119	68 118	syl5bir	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
120	119	ralimdva	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
121		ralbi	⊢ ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
122	120 121	syl6	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
123	67 122	syl5bir	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
124	123	imp	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
125	124	an4s	⊢ ( ( ( 𝜑 ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
126	125	anbi2d	⊢ ( ( ( 𝜑 ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
127	126	expr	⊢ ( ( ( 𝜑 ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) ) )
128	127	ralimdva	⊢ ( ( 𝜑 ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) ) )
129	128	expimpd	⊢ ( 𝜑 → ( ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) ) )
130		ralbi	⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
131	66 129 130	syl56	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) ) )
132	10 16 131	pm5.21ndd	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
133	1	homfeqbas	⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
134		eqid	⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
135		simpr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
136	70 71 134 74 135 135	homfeqval	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) )
137	133	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
138	74	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
139		simpr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) )
140		simpllr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
141	70 71 134 138 139 140	homfeqval	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) = ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) )
142	1	ad4antr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
143	2	ad4antr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )
144		simplr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) )
145		simp-4r	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
146		simpr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) )
147		simpllr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) )
148	70 71 72 73 142 143 144 145 145 146 147	comfeqval	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) )
149	148	eqeq1d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → ( ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ↔ ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ) )
150	141 149	raleqbidva	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ↔ ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ) )
151	70 71 134 138 140 139	homfeqval	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) )
152	1	ad4antr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
153	2	ad4antr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )
154		simp-4r	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
155		simplr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) )
156		simpllr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) )
157		simpr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) )
158	70 71 72 73 152 153 154 154 155 156 157	comfeqval	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) )
159	158	eqeq1d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ↔ ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) )
160	151 159	raleqbidva	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ↔ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) )
161	150 160	anbi12d	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ↔ ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ) )
162	137 161	raleqbidva	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ) )
163	136 162	rexeqbidva	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ↔ ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ) )
164	133	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
165	164	adantr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
166	74	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
167		simplr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) )
168	70 71 134 166 79 167	homfeqval	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) )
169		simpr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → 𝑧 ∈ ( Base ‘ 𝐶 ) )
170	70 71 134 166 167 169	homfeqval	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) )
171	170	adantr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) )
172		simpr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) )
173	70 71 72 73 75 77 80 82 89 85 172	comfeqval	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) )
174	70 71 134 166 79 169	homfeqval	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) )
175	174	ad2antrr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) )
176	173 175	eleq12d	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ) )
177	164	ad4antr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
178	75	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
179		simp-4r	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) → 𝑧 ∈ ( Base ‘ 𝐶 ) )
180		simpr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) → 𝑤 ∈ ( Base ‘ 𝐶 ) )
181	70 71 134 178 179 180	homfeqval	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) = ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) )
182	166	ad4antr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
183	2	ad7antr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )
184	167	ad4antr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) )
185	169	ad4antr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑧 ∈ ( Base ‘ 𝐶 ) )
186		simplr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑤 ∈ ( Base ‘ 𝐶 ) )
187		simpllr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) )
188		simpr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) )
189	70 71 72 73 182 183 184 185 186 187 188	comfeqval	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) = ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) )
190	189	oveq1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) )
191	79	ad4antr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
192		simp-4r	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) )
193	70 71 72 73 182 183 191 184 185 192 187	comfeqval	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) )
194	193	oveq2d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) )
195	190 194	eqeq12d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) )
196	181 195	raleqbidva	⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) )
197	177 196	raleqbidva	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) )
198	176 197	anbi12d	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) )
199	171 198	raleqbidva	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) )
200	168 199	raleqbidva	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) )
201	165 200	raleqbidva	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐷 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) )
202	164 201	raleqbidva	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ∀ 𝑧 ∈ ( Base ‘ 𝐷 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) )
203	163 202	anbi12d	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ∀ 𝑧 ∈ ( Base ‘ 𝐷 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) ) )
204	133 203	raleqbidva	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐷 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ∀ 𝑧 ∈ ( Base ‘ 𝐷 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) ) )
205	132 204	bitrd	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐷 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ∀ 𝑧 ∈ ( Base ‘ 𝐷 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) ) )
206	70 71 72	iscat	⊢ ( 𝐶 ∈ 𝑉 → ( 𝐶 ∈ Cat ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
207	3 206	syl	⊢ ( 𝜑 → ( 𝐶 ∈ Cat ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
208		eqid	⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
209	208 134 73	iscat	⊢ ( 𝐷 ∈ 𝑊 → ( 𝐷 ∈ Cat ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐷 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ∀ 𝑧 ∈ ( Base ‘ 𝐷 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) ) )
210	4 209	syl	⊢ ( 𝜑 → ( 𝐷 ∈ Cat ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐷 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ∀ 𝑧 ∈ ( Base ‘ 𝐷 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) ) )
211	205 207 210	3bitr4d	⊢ ( 𝜑 → ( 𝐶 ∈ Cat ↔ 𝐷 ∈ Cat ) )