Metamath Proof Explorer


Theorem rngcbas

Description: Set of objects 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 ( 𝜑𝑈𝑉 )
Assertion rngcbas ( 𝜑𝐵 = ( 𝑈 ∩ Rng ) )

Proof

Step Hyp Ref Expression
1 rngcbas.c 𝐶 = ( RngCat ‘ 𝑈 )
2 rngcbas.b 𝐵 = ( Base ‘ 𝐶 )
3 rngcbas.u ( 𝜑𝑈𝑉 )
4 eqidd ( 𝜑 → ( 𝑈 ∩ Rng ) = ( 𝑈 ∩ Rng ) )
5 eqidd ( 𝜑 → ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) = ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) )
6 1 3 4 5 rngcval ( 𝜑𝐶 = ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) )
7 6 fveq2d ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) ) )
8 2 a1i ( 𝜑𝐵 = ( Base ‘ 𝐶 ) )
9 eqid ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) = ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) )
10 eqid ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) = ( Base ‘ ( ExtStrCat ‘ 𝑈 ) )
11 fvexd ( 𝜑 → ( ExtStrCat ‘ 𝑈 ) ∈ V )
12 4 5 rnghmresfn ( 𝜑 → ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) Fn ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) )
13 inss1 ( 𝑈 ∩ Rng ) ⊆ 𝑈
14 eqid ( ExtStrCat ‘ 𝑈 ) = ( ExtStrCat ‘ 𝑈 )
15 14 3 estrcbas ( 𝜑𝑈 = ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) )
16 13 15 sseqtrid ( 𝜑 → ( 𝑈 ∩ Rng ) ⊆ ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) )
17 9 10 11 12 16 rescbas ( 𝜑 → ( 𝑈 ∩ Rng ) = ( Base ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) ) )
18 7 8 17 3eqtr4d ( 𝜑𝐵 = ( 𝑈 ∩ Rng ) )