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Theorem shftdm 10858
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.)
Hypothesis
Ref Expression
shftfval.1  |-  F  e. 
_V
Assertion
Ref Expression
shftdm  |-  ( A  e.  CC  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e. 
dom  F } )
Distinct variable groups:    x, A    x, F

Proof of Theorem shftdm
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 shftfval.1 . . . 4  |-  F  e. 
_V
21shftfval 10857 . . 3  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
32dmeqd 4844 . 2  |-  ( A  e.  CC  ->  dom  ( F  shift  A )  =  dom  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
4 19.42v 1918 . . . . 5  |-  ( E. y ( x  e.  CC  /\  ( x  -  A ) F y )  <->  ( x  e.  CC  /\  E. y
( x  -  A
) F y ) )
5 simpr 110 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  x  e.  CC )
6 simpl 109 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  A  e.  CC )
75, 6subcld 8293 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( x  -  A
)  e.  CC )
8 eldmg 4837 . . . . . . 7  |-  ( ( x  -  A )  e.  CC  ->  (
( x  -  A
)  e.  dom  F  <->  E. y ( x  -  A ) F y ) )
97, 8syl 14 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( ( x  -  A )  e.  dom  F  <->  E. y ( x  -  A ) F y ) )
109pm5.32da 452 . . . . 5  |-  ( A  e.  CC  ->  (
( x  e.  CC  /\  ( x  -  A
)  e.  dom  F
)  <->  ( x  e.  CC  /\  E. y
( x  -  A
) F y ) ) )
114, 10bitr4id 199 . . . 4  |-  ( A  e.  CC  ->  ( E. y ( x  e.  CC  /\  ( x  -  A ) F y )  <->  ( x  e.  CC  /\  ( x  -  A )  e. 
dom  F ) ) )
1211abbidv 2307 . . 3  |-  ( A  e.  CC  ->  { x  |  E. y ( x  e.  CC  /\  (
x  -  A ) F y ) }  =  { x  |  ( x  e.  CC  /\  ( x  -  A
)  e.  dom  F
) } )
13 dmopab 4853 . . 3  |-  dom  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  =  { x  |  E. y ( x  e.  CC  /\  ( x  -  A ) F y ) }
14 df-rab 2477 . . 3  |-  { x  e.  CC  |  ( x  -  A )  e. 
dom  F }  =  { x  |  (
x  e.  CC  /\  ( x  -  A
)  e.  dom  F
) }
1512, 13, 143eqtr4g 2247 . 2  |-  ( A  e.  CC  ->  dom  {
<. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  =  { x  e.  CC  |  ( x  -  A )  e.  dom  F } )
163, 15eqtrd 2222 1  |-  ( A  e.  CC  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e. 
dom  F } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   E.wex 1503    e. wcel 2160   {cab 2175   {crab 2472   _Vcvv 2752   class class class wbr 4018   {copab 4078   dom cdm 4641  (class class class)co 5892   CCcc 7834    - cmin 8153    shift cshi 10850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-resscn 7928  ax-1cn 7929  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-addcom 7936  ax-addass 7938  ax-distr 7940  ax-i2m1 7941  ax-0id 7944  ax-rnegex 7945  ax-cnre 7947
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-sub 8155  df-shft 10851
This theorem is referenced by:  shftfn  10860
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