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Metamath Proof Explorer

Theorem elringchom

Description: A morphism of unital rings is a function. (Contributed by AV, 14-Feb-2020)

Ref Expression
Hypotheses ringcbas.c 
|- C = ( RingCat ` U )
ringcbas.b 
|- B = ( Base ` C )
ringcbas.u 
|- ( ph -> U e. V )
ringchomfval.h 
|- H = ( Hom ` C )
ringchom.x 
|- ( ph -> X e. B )
ringchom.y 
|- ( ph -> Y e. B )
Assertion elringchom 
|- ( ph -> ( F e. ( X H Y ) -> F : ( Base ` X ) --> ( Base ` Y ) ) )

Proof

Step Hyp Ref Expression
1 ringcbas.c 
 |-  C = ( RingCat ` U )
2 ringcbas.b 
 |-  B = ( Base ` C )
3 ringcbas.u 
 |-  ( ph -> U e. V )
4 ringchomfval.h 
 |-  H = ( Hom ` C )
5 ringchom.x 
 |-  ( ph -> X e. B )
6 ringchom.y 
 |-  ( ph -> Y e. B )
7 1 2 3 4 5 6 ringchom 
 |-  ( ph -> ( X H Y ) = ( X RingHom Y ) )
8 7 eleq2d 
 |-  ( ph -> ( F e. ( X H Y ) <-> F e. ( X RingHom Y ) ) )
9 eqid 
 |-  ( Base ` X ) = ( Base ` X )
10 eqid 
 |-  ( Base ` Y ) = ( Base ` Y )
11 9 10 rhmf 
 |-  ( F e. ( X RingHom Y ) -> F : ( Base ` X ) --> ( Base ` Y ) )
12 8 11 biimtrdi 
 |-  ( ph -> ( F e. ( X H Y ) -> F : ( Base ` X ) --> ( Base ` Y ) ) )