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Metamath Proof Explorer

Theorem rngchomfval

Description: Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020) (Revised by AV, 8-Mar-2020)

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

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 1 2 3 rngcbas ⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Rng ) )
6 eqidd ⊢ ( 𝜑 → ( RngHom ↾ ( 𝐵 × 𝐵 ) ) = ( RngHom ↾ ( 𝐵 × 𝐵 ) ) )
7 1 3 5 6 rngcval ⊢ ( 𝜑 → 𝐶 = ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHom ↾ ( 𝐵 × 𝐵 ) ) ) )
8 7 fveq2d ⊢ ( 𝜑 → ( Hom ‘ 𝐶 ) = ( Hom ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHom ↾ ( 𝐵 × 𝐵 ) ) ) ) )
9 4 8 eqtrid ⊢ ( 𝜑 → 𝐻 = ( Hom ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHom ↾ ( 𝐵 × 𝐵 ) ) ) ) )
10 eqid ⊢ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHom ↾ ( 𝐵 × 𝐵 ) ) ) = ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHom ↾ ( 𝐵 × 𝐵 ) ) )
11 eqid ⊢ ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) = ( Base ‘ ( ExtStrCat ‘ 𝑈 ) )
12 fvexd ⊢ ( 𝜑 → ( ExtStrCat ‘ 𝑈 ) ∈ V )
13 5 6 rnghmresfn ⊢ ( 𝜑 → ( RngHom ↾ ( 𝐵 × 𝐵 ) ) Fn ( 𝐵 × 𝐵 ) )
14 inss1 ⊢ ( 𝑈 ∩ Rng ) ⊆ 𝑈
15 14 a1i ⊢ ( 𝜑 → ( 𝑈 ∩ Rng ) ⊆ 𝑈 )
16 eqid ⊢ ( ExtStrCat ‘ 𝑈 ) = ( ExtStrCat ‘ 𝑈 )
17 16 3 estrcbas ⊢ ( 𝜑 → 𝑈 = ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) )
18 17 eqcomd ⊢ ( 𝜑 → ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) = 𝑈 )
19 15 5 18 3sstr4d ⊢ ( 𝜑 → 𝐵 ⊆ ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) )
20 10 11 12 13 19 reschom ⊢ ( 𝜑 → ( RngHom ↾ ( 𝐵 × 𝐵 ) ) = ( Hom ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHom ↾ ( 𝐵 × 𝐵 ) ) ) ) )
21 9 20 eqtr4d ⊢ ( 𝜑 → 𝐻 = ( RngHom ↾ ( 𝐵 × 𝐵 ) ) )