Theorem comfeqd

Description: Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017)

	Ref	Expression
Hypotheses	comfeqd.1	⊢ ( 𝜑 → ( comp ‘ 𝐶 ) = ( comp ‘ 𝐷 ) )
	comfeqd.2	⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
Assertion	comfeqd	⊢ ( 𝜑 → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		comfeqd.1	⊢ ( 𝜑 → ( comp ‘ 𝐶 ) = ( comp ‘ 𝐷 ) )
2		comfeqd.2	⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
3	1	oveqd	⊢ ( 𝜑 → ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) = ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) )
4	3	oveqd	⊢ ( 𝜑 → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) )
5	4	ralrimivw	⊢ ( 𝜑 → ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) )
6	5	ralrimivw	⊢ ( 𝜑 → ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) )
7	6	ralrimivw	⊢ ( 𝜑 → ∀ 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 ) )
8	7	ralrimivw	⊢ ( 𝜑 → ∀ 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 ) )
9	8	ralrimivw	⊢ ( 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 ) )
10		eqid	⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 )
11		eqid	⊢ ( comp ‘ 𝐷 ) = ( comp ‘ 𝐷 )
12		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
13		eqidd	⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) )
14	2	homfeqbas	⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
15	10 11 12 13 14 2	comfeq	⊢ ( 𝜑 → ( ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) ↔ ∀ 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 ) ) )
16	9 15	mpbird	⊢ ( 𝜑 → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )