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Metamath Proof Explorer

Theorem estrcid

Description: The identity arrow in the category of extensible structures is the identity function of base sets. (Contributed by AV, 8-Mar-2020)

Ref Expression
Hypotheses estrccat.c 
|- C = ( ExtStrCat ` U )
estrcid.o 
|- .1. = ( Id ` C )
estrcid.u 
|- ( ph -> U e. V )
estrcid.x 
|- ( ph -> X e. U )
Assertion estrcid 
|- ( ph -> ( .1. ` X ) = ( _I |` ( Base ` X ) ) )

Proof

Step Hyp Ref Expression
1 estrccat.c 
 |-  C = ( ExtStrCat ` U )
2 estrcid.o 
 |-  .1. = ( Id ` C )
3 estrcid.u 
 |-  ( ph -> U e. V )
4 estrcid.x 
 |-  ( ph -> X e. U )
5 1 estrccatid 
 |-  ( U e. V -> ( C e. Cat /\ ( Id ` C ) = ( x e. U |-> ( _I |` ( Base ` x ) ) ) ) )
6 3 5 syl 
 |-  ( ph -> ( C e. Cat /\ ( Id ` C ) = ( x e. U |-> ( _I |` ( Base ` x ) ) ) ) )
7 6 simprd 
 |-  ( ph -> ( Id ` C ) = ( x e. U |-> ( _I |` ( Base ` x ) ) ) )
8 2 7 eqtrid 
 |-  ( ph -> .1. = ( x e. U |-> ( _I |` ( Base ` x ) ) ) )
9 fveq2 
 |-  ( x = X -> ( Base ` x ) = ( Base ` X ) )
10 9 reseq2d 
 |-  ( x = X -> ( _I |` ( Base ` x ) ) = ( _I |` ( Base ` X ) ) )
11 10 adantl 
 |-  ( ( ph /\ x = X ) -> ( _I |` ( Base ` x ) ) = ( _I |` ( Base ` X ) ) )
12 fvexd 
 |-  ( ph -> ( Base ` X ) e. _V )
13 12 resiexd 
 |-  ( ph -> ( _I |` ( Base ` X ) ) e. _V )
14 8 11 4 13 fvmptd 
 |-  ( ph -> ( .1. ` X ) = ( _I |` ( Base ` X ) ) )