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Metamath Proof Explorer

Theorem iorlid

Description: A magma right and left identity belongs to the underlying set of the operation. (Contributed by FL, 12-Dec-2009) (Revised by Mario Carneiro, 22-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypotheses iorlid.1 
|- X = ran G
iorlid.2 
|- U = ( GId ` G )
Assertion iorlid 
|- ( G e. ( Magma i^i ExId ) -> U e. X )

Proof

Step Hyp Ref Expression
1 iorlid.1 
 |-  X = ran G
2 iorlid.2 
 |-  U = ( GId ` G )
3 1 2 idrval 
 |-  ( G e. ( Magma i^i ExId ) -> U = ( iota_ u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) )
4 1 exidu1 
 |-  ( G e. ( Magma i^i ExId ) -> E! u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) )
5 riotacl 
 |-  ( E! u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) -> ( iota_ u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) e. X )
6 4 5 syl 
 |-  ( G e. ( Magma i^i ExId ) -> ( iota_ u e. X A. x e. X ( ( u G x ) = x /\ ( x G u ) = x ) ) e. X )
7 3 6 eqeltrd 
 |-  ( G e. ( Magma i^i ExId ) -> U e. X )