Theorem homfeqd

Description: If two structures have the same Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017)

	Ref	Expression
Hypotheses	homfeqd.1	⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
	homfeqd.2	⊢ ( 𝜑 → ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐷 ) )
Assertion	homfeqd	⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		homfeqd.1	⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
2		homfeqd.2	⊢ ( 𝜑 → ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐷 ) )
3	2	oveqd	⊢ ( 𝜑 → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) )
4	3	ralrimivw	⊢ ( 𝜑 → ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) )
5	4	ralrimivw	⊢ ( 𝜑 → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) )
6		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
7		eqid	⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
8		eqidd	⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) )
9	6 7 8 1	homfeq	⊢ ( 𝜑 → ( ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ) )
10	5 9	mpbird	⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )