Theorem elrngchom

Description: A morphism of non-unital rings is a function. (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	elrngchom	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) → 𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 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 5 6	rngchom	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 RngHomo 𝑌 ) )
8	7	eleq2d	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ↔ 𝐹 ∈ ( 𝑋 RngHomo 𝑌 ) ) )
9		eqid	⊢ ( Base ‘ 𝑋 ) = ( Base ‘ 𝑋 )
10		eqid	⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 )
11	9 10	rnghmf	⊢ ( 𝐹 ∈ ( 𝑋 RngHomo 𝑌 ) → 𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) )
12	8 11	syl6bi	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) → 𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ) )