Metamath Proof Explorer


Theorem subrfn

Description: Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012)

Ref Expression
Assertion subrfn
|- ( ( A e. C /\ B e. D ) -> ( A -r B ) Fn RR )

Proof

Step Hyp Ref Expression
1 ovex
 |-  ( ( A ` x ) - ( B ` x ) ) e. _V
2 eqid
 |-  ( x e. RR |-> ( ( A ` x ) - ( B ` x ) ) ) = ( x e. RR |-> ( ( A ` x ) - ( B ` x ) ) )
3 1 2 fnmpti
 |-  ( x e. RR |-> ( ( A ` x ) - ( B ` x ) ) ) Fn RR
4 subrval
 |-  ( ( A e. C /\ B e. D ) -> ( A -r B ) = ( x e. RR |-> ( ( A ` x ) - ( B ` x ) ) ) )
5 4 fneq1d
 |-  ( ( A e. C /\ B e. D ) -> ( ( A -r B ) Fn RR <-> ( x e. RR |-> ( ( A ` x ) - ( B ` x ) ) ) Fn RR ) )
6 3 5 mpbiri
 |-  ( ( A e. C /\ B e. D ) -> ( A -r B ) Fn RR )