Metamath Proof Explorer


Theorem fndmdifeq0

Description: The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015)

Ref Expression
Assertion fndmdifeq0
|- ( ( F Fn A /\ G Fn A ) -> ( dom ( F \ G ) = (/) <-> F = G ) )

Proof

Step Hyp Ref Expression
1 fndmdif
 |-  ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )
2 1 eqeq1d
 |-  ( ( F Fn A /\ G Fn A ) -> ( dom ( F \ G ) = (/) <-> { x e. A | ( F ` x ) =/= ( G ` x ) } = (/) ) )
3 eqfnfv
 |-  ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
4 rabeq0
 |-  ( { x e. A | ( F ` x ) =/= ( G ` x ) } = (/) <-> A. x e. A -. ( F ` x ) =/= ( G ` x ) )
5 nne
 |-  ( -. ( F ` x ) =/= ( G ` x ) <-> ( F ` x ) = ( G ` x ) )
6 5 ralbii
 |-  ( A. x e. A -. ( F ` x ) =/= ( G ` x ) <-> A. x e. A ( F ` x ) = ( G ` x ) )
7 4 6 bitri
 |-  ( { x e. A | ( F ` x ) =/= ( G ` x ) } = (/) <-> A. x e. A ( F ` x ) = ( G ` x ) )
8 3 7 syl6rbbr
 |-  ( ( F Fn A /\ G Fn A ) -> ( { x e. A | ( F ` x ) =/= ( G ` x ) } = (/) <-> F = G ) )
9 2 8 bitrd
 |-  ( ( F Fn A /\ G Fn A ) -> ( dom ( F \ G ) = (/) <-> F = G ) )