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Metamath Proof Explorer

Theorem mndtcid

Description: The identity morphism, or identity arrow, of the category built from a monoid is the identity element of the monoid. (Contributed by Zhi Wang, 22-Sep-2024)

Ref Expression
Hypotheses mndtccat.c 
|- ( ph -> C = ( MndToCat ` M ) )
mndtccat.m 
|- ( ph -> M e. Mnd )
mndtcid.b 
|- ( ph -> B = ( Base ` C ) )
mndtcid.x 
|- ( ph -> X e. B )
mndtcid.i 
|- ( ph -> .1. = ( Id ` C ) )
Assertion mndtcid 
|- ( ph -> ( .1. ` X ) = ( 0g ` M ) )

Proof

Step Hyp Ref Expression
1 mndtccat.c 
 |-  ( ph -> C = ( MndToCat ` M ) )
2 mndtccat.m 
 |-  ( ph -> M e. Mnd )
3 mndtcid.b 
 |-  ( ph -> B = ( Base ` C ) )
4 mndtcid.x 
 |-  ( ph -> X e. B )
5 mndtcid.i 
 |-  ( ph -> .1. = ( Id ` C ) )
6 1 2 mndtccatid 
 |-  ( ph -> ( C e. Cat /\ ( Id ` C ) = ( x e. ( Base ` C ) |-> ( 0g ` M ) ) ) )
7 6 simprd 
 |-  ( ph -> ( Id ` C ) = ( x e. ( Base ` C ) |-> ( 0g ` M ) ) )
8 5 7 eqtrd 
 |-  ( ph -> .1. = ( x e. ( Base ` C ) |-> ( 0g ` M ) ) )
9 eqidd 
 |-  ( ( ph /\ x = X ) -> ( 0g ` M ) = ( 0g ` M ) )
10 4 3 eleqtrd 
 |-  ( ph -> X e. ( Base ` C ) )
11 fvexd 
 |-  ( ph -> ( 0g ` M ) e. _V )
12 8 9 10 11 fvmptd 
 |-  ( ph -> ( .1. ` X ) = ( 0g ` M ) )