Theorem rngchom

Description: Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020)

	Ref	Expression
Hypotheses	rngcbas.c	⊢ 𝐶 = ( RngCat ‘ 𝑈 )
	rngcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	rngcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
	rngchomfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	rngchom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	rngchom.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	rngchom	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 RngHom 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		rngcbas.c	⊢ 𝐶 = ( RngCat ‘ 𝑈 )
2		rngcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		rngcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
4		rngchomfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
5		rngchom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		rngchom.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7	1 2 3 4	rngchomfval	⊢ ( 𝜑 → 𝐻 = ( RngHom ↾ ( 𝐵 × 𝐵 ) ) )
8	7	oveqd	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 ( RngHom ↾ ( 𝐵 × 𝐵 ) ) 𝑌 ) )
9	5 6	ovresd	⊢ ( 𝜑 → ( 𝑋 ( RngHom ↾ ( 𝐵 × 𝐵 ) ) 𝑌 ) = ( 𝑋 RngHom 𝑌 ) )
10	8 9	eqtrd	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 RngHom 𝑌 ) )