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Metamath Proof Explorer

Theorem setcid

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

Ref Expression
Hypotheses setccat.c 
|- C = ( SetCat ` U )
setcid.o 
|- .1. = ( Id ` C )
setcid.u 
|- ( ph -> U e. V )
setcid.x 
|- ( ph -> X e. U )
Assertion setcid 
|- ( ph -> ( .1. ` X ) = ( _I |` X ) )

Proof

Step Hyp Ref Expression
1 setccat.c 
 |-  C = ( SetCat ` U )
2 setcid.o 
 |-  .1. = ( Id ` C )
3 setcid.u 
 |-  ( ph -> U e. V )
4 setcid.x 
 |-  ( ph -> X e. U )
5 1 setccatid 
 |-  ( U e. V -> ( C e. Cat /\ ( Id ` C ) = ( x e. U |-> ( _I |` x ) ) ) )
6 3 5 syl 
 |-  ( ph -> ( C e. Cat /\ ( Id ` C ) = ( x e. U |-> ( _I |` x ) ) ) )
7 6 simprd 
 |-  ( ph -> ( Id ` C ) = ( x e. U |-> ( _I |` x ) ) )
8 2 7 eqtrid 
 |-  ( ph -> .1. = ( x e. U |-> ( _I |` x ) ) )
9 simpr 
 |-  ( ( ph /\ x = X ) -> x = X )
10 9 reseq2d 
 |-  ( ( ph /\ x = X ) -> ( _I |` x ) = ( _I |` X ) )
11 4 resiexd 
 |-  ( ph -> ( _I |` X ) e. _V )
12 8 10 4 11 fvmptd 
 |-  ( ph -> ( .1. ` X ) = ( _I |` X ) )