Metamath Proof Explorer


Theorem shftdm

Description: Domain of a relation shifted by A . The set on the right is more commonly notated as ( dom F + A ) (meaning add A to every element of dom F ). (Contributed by Mario Carneiro, 3-Nov-2013)

Ref Expression
Hypothesis shftfval.1
|- F e. _V
Assertion shftdm
|- ( A e. CC -> dom ( F shift A ) = { x e. CC | ( x - A ) e. dom F } )

Proof

Step Hyp Ref Expression
1 shftfval.1
 |-  F e. _V
2 1 shftfval
 |-  ( A e. CC -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
3 2 dmeqd
 |-  ( A e. CC -> dom ( F shift A ) = dom { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
4 19.42v
 |-  ( E. y ( x e. CC /\ ( x - A ) F y ) <-> ( x e. CC /\ E. y ( x - A ) F y ) )
5 ovex
 |-  ( x - A ) e. _V
6 5 eldm
 |-  ( ( x - A ) e. dom F <-> E. y ( x - A ) F y )
7 6 anbi2i
 |-  ( ( x e. CC /\ ( x - A ) e. dom F ) <-> ( x e. CC /\ E. y ( x - A ) F y ) )
8 4 7 bitr4i
 |-  ( E. y ( x e. CC /\ ( x - A ) F y ) <-> ( x e. CC /\ ( x - A ) e. dom F ) )
9 8 abbii
 |-  { x | E. y ( x e. CC /\ ( x - A ) F y ) } = { x | ( x e. CC /\ ( x - A ) e. dom F ) }
10 dmopab
 |-  dom { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } = { x | E. y ( x e. CC /\ ( x - A ) F y ) }
11 df-rab
 |-  { x e. CC | ( x - A ) e. dom F } = { x | ( x e. CC /\ ( x - A ) e. dom F ) }
12 9 10 11 3eqtr4i
 |-  dom { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } = { x e. CC | ( x - A ) e. dom F }
13 3 12 eqtrdi
 |-  ( A e. CC -> dom ( F shift A ) = { x e. CC | ( x - A ) e. dom F } )