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Metamath Proof Explorer

Theorem catccofval

Description: Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017)

Ref Expression
Hypotheses catcbas.c 
|- C = ( CatCat ` U )
catcbas.b 
|- B = ( Base ` C )
catcbas.u 
|- ( ph -> U e. V )
catcco.o 
|- .x. = ( comp ` C )
Assertion catccofval 
|- ( ph -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) )

Proof

Step Hyp Ref Expression
1 catcbas.c 
 |-  C = ( CatCat ` U )
2 catcbas.b 
 |-  B = ( Base ` C )
3 catcbas.u 
 |-  ( ph -> U e. V )
4 catcco.o 
 |-  .x. = ( comp ` C )
5 1 2 3 catcbas 
 |-  ( ph -> B = ( U i^i Cat ) )
6 eqid 
 |-  ( Hom ` C ) = ( Hom ` C )
7 1 2 3 6 catchomfval 
 |-  ( ph -> ( Hom ` C ) = ( x e. B , y e. B |-> ( x Func y ) ) )
8 eqidd 
 |-  ( ph -> ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) )
9 1 3 5 7 8 catcval 
 |-  ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , ( Hom ` C ) >. , <. ( comp ` ndx ) , ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } )
10 9 fveq2d 
 |-  ( ph -> ( comp ` C ) = ( comp ` { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , ( Hom ` C ) >. , <. ( comp ` ndx ) , ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } ) )
11 2 fvexi 
 |-  B e. _V
12 11 11 xpex 
 |-  ( B X. B ) e. _V
13 12 11 mpoex 
 |-  ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) e. _V
14 catstr 
 |-  { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , ( Hom ` C ) >. , <. ( comp ` ndx ) , ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } Struct <. 1 , ; 1 5 >.
15 ccoid 
 |-  comp = Slot ( comp ` ndx )
16 snsstp3 
 |-  { <. ( comp ` ndx ) , ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } C_ { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , ( Hom ` C ) >. , <. ( comp ` ndx ) , ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. }
17 14 15 16 strfv 
 |-  ( ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) e. _V -> ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) = ( comp ` { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , ( Hom ` C ) >. , <. ( comp ` ndx ) , ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } ) )
18 13 17 ax-mp 
 |-  ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) = ( comp ` { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , ( Hom ` C ) >. , <. ( comp ` ndx ) , ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } )
19 10 4 18 3eqtr4g 
 |-  ( ph -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) )