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Metamath Proof Explorer

Theorem catidcl

Description: Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017)

Ref Expression
Hypotheses catidcl.b 
|- B = ( Base ` C )
catidcl.h 
|- H = ( Hom ` C )
catidcl.i 
|- .1. = ( Id ` C )
catidcl.c 
|- ( ph -> C e. Cat )
catidcl.x 
|- ( ph -> X e. B )
Assertion catidcl 
|- ( ph -> ( .1. ` X ) e. ( X H X ) )

Proof

Step Hyp Ref Expression
1 catidcl.b 
 |-  B = ( Base ` C )
2 catidcl.h 
 |-  H = ( Hom ` C )
3 catidcl.i 
 |-  .1. = ( Id ` C )
4 catidcl.c 
 |-  ( ph -> C e. Cat )
5 catidcl.x 
 |-  ( ph -> X e. B )
6 eqid 
 |-  ( comp ` C ) = ( comp ` C )
7 1 2 6 4 3 5 cidval 
 |-  ( ph -> ( .1. ` X ) = ( iota_ g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. ( comp ` C ) X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. ( comp ` C ) y ) g ) = f ) ) )
8 1 2 6 4 5 catideu 
 |-  ( ph -> E! g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. ( comp ` C ) X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. ( comp ` C ) y ) g ) = f ) )
9 riotacl 
 |-  ( E! g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. ( comp ` C ) X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. ( comp ` C ) y ) g ) = f ) -> ( iota_ g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. ( comp ` C ) X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. ( comp ` C ) y ) g ) = f ) ) e. ( X H X ) )
10 8 9 syl 
 |-  ( ph -> ( iota_ g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. ( comp ` C ) X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. ( comp ` C ) y ) g ) = f ) ) e. ( X H X ) )
11 7 10 eqeltrd 
 |-  ( ph -> ( .1. ` X ) e. ( X H X ) )