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Theorem shftdm 11814
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 11813 . . 3  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
32dmeqd 5013 . 2  |-  ( A  e.  CC  ->  dom  ( F  shift  A )  =  dom  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
4 19.42v 1917 . . . . 5  |-  ( E. y ( x  e.  CC  /\  ( x  -  A ) F y )  <->  ( x  e.  CC  /\  E. y
( x  -  A
) F y ) )
5 ovex 6046 . . . . . . 7  |-  ( x  -  A )  e. 
_V
65eldm 5008 . . . . . 6  |-  ( ( x  -  A )  e.  dom  F  <->  E. y
( x  -  A
) F y )
76anbi2i 676 . . . . 5  |-  ( ( x  e.  CC  /\  ( x  -  A
)  e.  dom  F
)  <->  ( x  e.  CC  /\  E. y
( x  -  A
) F y ) )
84, 7bitr4i 244 . . . 4  |-  ( E. y ( x  e.  CC  /\  ( x  -  A ) F y )  <->  ( x  e.  CC  /\  ( x  -  A )  e. 
dom  F ) )
98abbii 2500 . . 3  |-  { x  |  E. y ( x  e.  CC  /\  (
x  -  A ) F y ) }  =  { x  |  ( x  e.  CC  /\  ( x  -  A
)  e.  dom  F
) }
10 dmopab 5021 . . 3  |-  dom  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  =  { x  |  E. y ( x  e.  CC  /\  ( x  -  A ) F y ) }
11 df-rab 2659 . . 3  |-  { x  e.  CC  |  ( x  -  A )  e. 
dom  F }  =  { x  |  (
x  e.  CC  /\  ( x  -  A
)  e.  dom  F
) }
129, 10, 113eqtr4i 2418 . 2  |-  dom  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  =  { x  e.  CC  |  ( x  -  A )  e.  dom  F }
133, 12syl6eq 2436 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 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2374   {crab 2654   _Vcvv 2900   class class class wbr 4154   {copab 4207   dom cdm 4819  (class class class)co 6021   CCcc 8922    - cmin 9224    shift cshi 11809
This theorem is referenced by:  shftfn  11816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-ltxr 9059  df-sub 9226  df-shft 11810
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