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Metamath Proof Explorer

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 RngHom 𝑌 ) )
8 7 eleq2d ⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ↔ 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) ) )
9 eqid ⊢ ( Base ‘ 𝑋 ) = ( Base ‘ 𝑋 )
10 eqid ⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 )
11 9 10 rnghmf ⊢ ( 𝐹 ∈ ( 𝑋 RngHom 𝑌 ) → 𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) )
12 8 11 biimtrdi ⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) → 𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ) )