Metamath Proof Explorer

Theorem rngcbasALTV

Description: Set of objects of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020)

Ref Expression
Hypotheses rngcbasALTV.c 
|- C = ( RngCatALTV ` U )
rngcbasALTV.b 
|- B = ( Base ` C )
rngcbasALTV.u 
|- ( ph -> U e. V )
Assertion rngcbasALTV 
|- ( ph -> B = ( U i^i Rng ) )

Proof

Step Hyp Ref Expression
1 rngcbasALTV.c 
 |-  C = ( RngCatALTV ` U )
2 rngcbasALTV.b 
 |-  B = ( Base ` C )
3 rngcbasALTV.u 
 |-  ( ph -> U e. V )
4 eqidd 
 |-  ( ph -> ( U i^i Rng ) = ( U i^i Rng ) )
5 eqidd 
 |-  ( ph -> ( x e. ( U i^i Rng ) , y e. ( U i^i Rng ) |-> ( x RngHom y ) ) = ( x e. ( U i^i Rng ) , y e. ( U i^i Rng ) |-> ( x RngHom y ) ) )
6 eqidd 
 |-  ( ph -> ( v e. ( ( U i^i Rng ) X. ( U i^i Rng ) ) , z e. ( U i^i Rng ) |-> ( f e. ( ( 2nd ` v ) RngHom z ) , g e. ( ( 1st ` v ) RngHom ( 2nd ` v ) ) |-> ( f o. g ) ) ) = ( v e. ( ( U i^i Rng ) X. ( U i^i Rng ) ) , z e. ( U i^i Rng ) |-> ( f e. ( ( 2nd ` v ) RngHom z ) , g e. ( ( 1st ` v ) RngHom ( 2nd ` v ) ) |-> ( f o. g ) ) ) )
7 1 3 4 5 6 rngcvalALTV 
 |-  ( ph -> C = { <. ( Base ` ndx ) , ( U i^i Rng ) >. , <. ( Hom ` ndx ) , ( x e. ( U i^i Rng ) , y e. ( U i^i Rng ) |-> ( x RngHom y ) ) >. , <. ( comp ` ndx ) , ( v e. ( ( U i^i Rng ) X. ( U i^i Rng ) ) , z e. ( U i^i Rng ) |-> ( f e. ( ( 2nd ` v ) RngHom z ) , g e. ( ( 1st ` v ) RngHom ( 2nd ` v ) ) |-> ( f o. g ) ) ) >. } )
8 catstr 
 |-  { <. ( Base ` ndx ) , ( U i^i Rng ) >. , <. ( Hom ` ndx ) , ( x e. ( U i^i Rng ) , y e. ( U i^i Rng ) |-> ( x RngHom y ) ) >. , <. ( comp ` ndx ) , ( v e. ( ( U i^i Rng ) X. ( U i^i Rng ) ) , z e. ( U i^i Rng ) |-> ( f e. ( ( 2nd ` v ) RngHom z ) , g e. ( ( 1st ` v ) RngHom ( 2nd ` v ) ) |-> ( f o. g ) ) ) >. } Struct <. 1 , ; 1 5 >.
9 baseid 
 |-  Base = Slot ( Base ` ndx )
10 snsstp1 
 |-  { <. ( Base ` ndx ) , ( U i^i Rng ) >. } C_ { <. ( Base ` ndx ) , ( U i^i Rng ) >. , <. ( Hom ` ndx ) , ( x e. ( U i^i Rng ) , y e. ( U i^i Rng ) |-> ( x RngHom y ) ) >. , <. ( comp ` ndx ) , ( v e. ( ( U i^i Rng ) X. ( U i^i Rng ) ) , z e. ( U i^i Rng ) |-> ( f e. ( ( 2nd ` v ) RngHom z ) , g e. ( ( 1st ` v ) RngHom ( 2nd ` v ) ) |-> ( f o. g ) ) ) >. }
11 inex1g 
 |-  ( U e. V -> ( U i^i Rng ) e. _V )
12 3 11 syl 
 |-  ( ph -> ( U i^i Rng ) e. _V )
13 7 8 9 10 12 2 strfv3 
 |-  ( ph -> B = ( U i^i Rng ) )