Metamath Proof Explorer

Theorem estrcbas

Description: Set of objects of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020)

Ref Expression
Hypotheses estrcbas.c 
|- C = ( ExtStrCat ` U )
estrcbas.u 
|- ( ph -> U e. V )
Assertion estrcbas 
|- ( ph -> U = ( Base ` C ) )

Proof

Step Hyp Ref Expression
1 estrcbas.c 
 |-  C = ( ExtStrCat ` U )
2 estrcbas.u 
 |-  ( ph -> U e. V )
3 catstr 
 |-  { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) >. , <. ( comp ` ndx ) , ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. } Struct <. 1 , ; 1 5 >.
4 baseid 
 |-  Base = Slot ( Base ` ndx )
5 snsstp1 
 |-  { <. ( Base ` ndx ) , U >. } C_ { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) >. , <. ( comp ` ndx ) , ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. }
6 3 4 5 strfv 
 |-  ( U e. V -> U = ( Base ` { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) >. , <. ( comp ` ndx ) , ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. } ) )
7 2 6 syl 
 |-  ( ph -> U = ( Base ` { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) >. , <. ( comp ` ndx ) , ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. } ) )
8 eqidd 
 |-  ( ph -> ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) = ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )
9 eqidd 
 |-  ( ph -> ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) = ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) )
10 1 2 8 9 estrcval 
 |-  ( ph -> C = { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) >. , <. ( comp ` ndx ) , ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. } )
11 10 fveq2d 
 |-  ( ph -> ( Base ` C ) = ( Base ` { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) >. , <. ( comp ` ndx ) , ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. } ) )
12 7 11 eqtr4d 
 |-  ( ph -> U = ( Base ` C ) )