Theorem elringchom

Description: A morphism of unital rings is a function. (Contributed by AV, 14-Feb-2020)

	Ref	Expression
Hypotheses	ringcbas.c	⊢ 𝐶 = ( RingCat ‘ 𝑈 )
	ringcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	ringcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
	ringchomfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	ringchom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ringchom.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	elringchom	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) → 𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ringcbas.c	⊢ 𝐶 = ( RingCat ‘ 𝑈 )
2		ringcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		ringcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
4		ringchomfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
5		ringchom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		ringchom.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7	1 2 3 4 5 6	ringchom	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 RingHom 𝑌 ) )
8	7	eleq2d	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ↔ 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ) )
9		eqid	⊢ ( Base ‘ 𝑋 ) = ( Base ‘ 𝑋 )
10		eqid	⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 )
11	9 10	rhmf	⊢ ( 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) → 𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) )
12	8 11	biimtrdi	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) → 𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ) )