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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 } )