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Metamath Proof Explorer

Theorem rnghmfn

Description: The mapping of two non-unital rings to the non-unital ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020)

Ref Expression
Assertion rnghmfn 
|- RngHom Fn ( Rng X. Rng )

Proof

Step Hyp Ref Expression
1 df-rnghm 
 |-  RngHom = ( r e. Rng , s e. Rng |-> [_ ( Base ` r ) / v ]_ [_ ( Base ` s ) / w ]_ { f e. ( w ^m v ) | A. x e. v A. y e. v ( ( f ` ( x ( +g ` r ) y ) ) = ( ( f ` x ) ( +g ` s ) ( f ` y ) ) /\ ( f ` ( x ( .r ` r ) y ) ) = ( ( f ` x ) ( .r ` s ) ( f ` y ) ) ) } )
2 ovex 
 |-  ( w ^m v ) e. _V
3 2 rabex 
 |-  { f e. ( w ^m v ) | A. x e. v A. y e. v ( ( f ` ( x ( +g ` r ) y ) ) = ( ( f ` x ) ( +g ` s ) ( f ` y ) ) /\ ( f ` ( x ( .r ` r ) y ) ) = ( ( f ` x ) ( .r ` s ) ( f ` y ) ) ) } e. _V
4 3 csbex 
 |-  [_ ( Base ` s ) / w ]_ { f e. ( w ^m v ) | A. x e. v A. y e. v ( ( f ` ( x ( +g ` r ) y ) ) = ( ( f ` x ) ( +g ` s ) ( f ` y ) ) /\ ( f ` ( x ( .r ` r ) y ) ) = ( ( f ` x ) ( .r ` s ) ( f ` y ) ) ) } e. _V
5 4 csbex 
 |-  [_ ( Base ` r ) / v ]_ [_ ( Base ` s ) / w ]_ { f e. ( w ^m v ) | A. x e. v A. y e. v ( ( f ` ( x ( +g ` r ) y ) ) = ( ( f ` x ) ( +g ` s ) ( f ` y ) ) /\ ( f ` ( x ( .r ` r ) y ) ) = ( ( f ` x ) ( .r ` s ) ( f ` y ) ) ) } e. _V
6 1 5 fnmpoi 
 |-  RngHom Fn ( Rng X. Rng )