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Metamath Proof Explorer

Theorem catcid

Description: The identity arrow in the category of categories is the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017)

Ref Expression
Hypotheses catccatid.c 
|- C = ( CatCat ` U )
catccatid.b 
|- B = ( Base ` C )
catcid.o 
|- .1. = ( Id ` C )
catcid.i 
|- I = ( idFunc ` X )
catcid.u 
|- ( ph -> U e. V )
catcid.x 
|- ( ph -> X e. B )
Assertion catcid 
|- ( ph -> ( .1. ` X ) = I )

Proof

Step Hyp Ref Expression
1 catccatid.c 
 |-  C = ( CatCat ` U )
2 catccatid.b 
 |-  B = ( Base ` C )
3 catcid.o 
 |-  .1. = ( Id ` C )
4 catcid.i 
 |-  I = ( idFunc ` X )
5 catcid.u 
 |-  ( ph -> U e. V )
6 catcid.x 
 |-  ( ph -> X e. B )
7 1 2 catccatid 
 |-  ( U e. V -> ( C e. Cat /\ ( Id ` C ) = ( x e. B |-> ( idFunc ` x ) ) ) )
8 5 7 syl 
 |-  ( ph -> ( C e. Cat /\ ( Id ` C ) = ( x e. B |-> ( idFunc ` x ) ) ) )
9 8 simprd 
 |-  ( ph -> ( Id ` C ) = ( x e. B |-> ( idFunc ` x ) ) )
10 3 9 eqtrid 
 |-  ( ph -> .1. = ( x e. B |-> ( idFunc ` x ) ) )
11 simpr 
 |-  ( ( ph /\ x = X ) -> x = X )
12 11 fveq2d 
 |-  ( ( ph /\ x = X ) -> ( idFunc ` x ) = ( idFunc ` X ) )
13 fvexd 
 |-  ( ph -> ( idFunc ` X ) e. _V )
14 10 12 6 13 fvmptd 
 |-  ( ph -> ( .1. ` X ) = ( idFunc ` X ) )
15 14 4 eqtr4di 
 |-  ( ph -> ( .1. ` X ) = I )