Metamath Proof Explorer


Theorem rngcidALTV

Description: The identity arrow in the category of non-unital rings is the identity function. (Contributed by AV, 27-Feb-2020) (New usage is discouraged.)

Ref Expression
Hypotheses rngccatALTV.c
|- C = ( RngCatALTV ` U )
rngcidALTV.b
|- B = ( Base ` C )
rngcidALTV.o
|- .1. = ( Id ` C )
rngcidALTV.u
|- ( ph -> U e. V )
rngcidALTV.x
|- ( ph -> X e. B )
rngcidALTV.s
|- S = ( Base ` X )
Assertion rngcidALTV
|- ( ph -> ( .1. ` X ) = ( _I |` S ) )

Proof

Step Hyp Ref Expression
1 rngccatALTV.c
 |-  C = ( RngCatALTV ` U )
2 rngcidALTV.b
 |-  B = ( Base ` C )
3 rngcidALTV.o
 |-  .1. = ( Id ` C )
4 rngcidALTV.u
 |-  ( ph -> U e. V )
5 rngcidALTV.x
 |-  ( ph -> X e. B )
6 rngcidALTV.s
 |-  S = ( Base ` X )
7 1 2 rngccatidALTV
 |-  ( U e. V -> ( C e. Cat /\ ( Id ` C ) = ( x e. B |-> ( _I |` ( Base ` x ) ) ) ) )
8 4 7 syl
 |-  ( ph -> ( C e. Cat /\ ( Id ` C ) = ( x e. B |-> ( _I |` ( Base ` x ) ) ) ) )
9 8 simprd
 |-  ( ph -> ( Id ` C ) = ( x e. B |-> ( _I |` ( Base ` x ) ) ) )
10 3 9 syl5eq
 |-  ( ph -> .1. = ( x e. B |-> ( _I |` ( Base ` x ) ) ) )
11 fveq2
 |-  ( x = X -> ( Base ` x ) = ( Base ` X ) )
12 11 adantl
 |-  ( ( ph /\ x = X ) -> ( Base ` x ) = ( Base ` X ) )
13 12 reseq2d
 |-  ( ( ph /\ x = X ) -> ( _I |` ( Base ` x ) ) = ( _I |` ( Base ` X ) ) )
14 fvex
 |-  ( Base ` X ) e. _V
15 resiexg
 |-  ( ( Base ` X ) e. _V -> ( _I |` ( Base ` X ) ) e. _V )
16 14 15 mp1i
 |-  ( ph -> ( _I |` ( Base ` X ) ) e. _V )
17 10 13 5 16 fvmptd
 |-  ( ph -> ( .1. ` X ) = ( _I |` ( Base ` X ) ) )
18 6 reseq2i
 |-  ( _I |` S ) = ( _I |` ( Base ` X ) )
19 17 18 eqtr4di
 |-  ( ph -> ( .1. ` X ) = ( _I |` S ) )