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Theorem shftdm 10387
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 10386 . . 3  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
32dmeqd 4669 . 2  |-  ( A  e.  CC  ->  dom  ( F  shift  A )  =  dom  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
4 simpr 109 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  x  e.  CC )
5 simpl 108 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  A  e.  CC )
64, 5subcld 7890 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( x  -  A
)  e.  CC )
7 eldmg 4662 . . . . . . 7  |-  ( ( x  -  A )  e.  CC  ->  (
( x  -  A
)  e.  dom  F  <->  E. y ( x  -  A ) F y ) )
86, 7syl 14 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( ( x  -  A )  e.  dom  F  <->  E. y ( x  -  A ) F y ) )
98pm5.32da 441 . . . . 5  |-  ( A  e.  CC  ->  (
( x  e.  CC  /\  ( x  -  A
)  e.  dom  F
)  <->  ( x  e.  CC  /\  E. y
( x  -  A
) F y ) ) )
10 19.42v 1841 . . . . 5  |-  ( E. y ( x  e.  CC  /\  ( x  -  A ) F y )  <->  ( x  e.  CC  /\  E. y
( x  -  A
) F y ) )
119, 10syl6rbbr 198 . . . 4  |-  ( A  e.  CC  ->  ( E. y ( x  e.  CC  /\  ( x  -  A ) F y )  <->  ( x  e.  CC  /\  ( x  -  A )  e. 
dom  F ) ) )
1211abbidv 2212 . . 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 4678 . . 3  |-  dom  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  =  { x  |  E. y ( x  e.  CC  /\  ( x  -  A ) F y ) }
14 df-rab 2379 . . 3  |-  { x  e.  CC  |  ( x  -  A )  e. 
dom  F }  =  { x  |  (
x  e.  CC  /\  ( x  -  A
)  e.  dom  F
) }
1512, 13, 143eqtr4g 2152 . 2  |-  ( A  e.  CC  ->  dom  {
<. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  =  { x  e.  CC  |  ( x  -  A )  e.  dom  F } )
163, 15eqtrd 2127 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 103    <-> wb 104    = wceq 1296   E.wex 1433    e. wcel 1445   {cab 2081   {crab 2374   _Vcvv 2633   class class class wbr 3867   {copab 3920   dom cdm 4467  (class class class)co 5690   CCcc 7445    - cmin 7750    shift cshi 10379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-resscn 7534  ax-1cn 7535  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-addcom 7542  ax-addass 7544  ax-distr 7546  ax-i2m1 7547  ax-0id 7550  ax-rnegex 7551  ax-cnre 7553
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-sub 7752  df-shft 10380
This theorem is referenced by:  shftfn  10389
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