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Theorem shftdm 13211
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 13210 . . 3  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
32dmeqd 5042 . 2  |-  ( A  e.  CC  ->  dom  ( F  shift  A )  =  dom  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
4 19.42v 1842 . . . . 5  |-  ( E. y ( x  e.  CC  /\  ( x  -  A ) F y )  <->  ( x  e.  CC  /\  E. y
( x  -  A
) F y ) )
5 ovex 6336 . . . . . . 7  |-  ( x  -  A )  e. 
_V
65eldm 5037 . . . . . 6  |-  ( ( x  -  A )  e.  dom  F  <->  E. y
( x  -  A
) F y )
76anbi2i 708 . . . . 5  |-  ( ( x  e.  CC  /\  ( x  -  A
)  e.  dom  F
)  <->  ( x  e.  CC  /\  E. y
( x  -  A
) F y ) )
84, 7bitr4i 260 . . . 4  |-  ( E. y ( x  e.  CC  /\  ( x  -  A ) F y )  <->  ( x  e.  CC  /\  ( x  -  A )  e. 
dom  F ) )
98abbii 2587 . . 3  |-  { x  |  E. y ( x  e.  CC  /\  (
x  -  A ) F y ) }  =  { x  |  ( x  e.  CC  /\  ( x  -  A
)  e.  dom  F
) }
10 dmopab 5051 . . 3  |-  dom  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  =  { x  |  E. y ( x  e.  CC  /\  ( x  -  A ) F y ) }
11 df-rab 2765 . . 3  |-  { x  e.  CC  |  ( x  -  A )  e. 
dom  F }  =  { x  |  (
x  e.  CC  /\  ( x  -  A
)  e.  dom  F
) }
129, 10, 113eqtr4i 2503 . 2  |-  dom  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  =  { x  e.  CC  |  ( x  -  A )  e.  dom  F }
133, 12syl6eq 2521 1  |-  ( A  e.  CC  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e. 
dom  F } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457   {crab 2760   _Vcvv 3031   class class class wbr 4395   {copab 4453   dom cdm 4839  (class class class)co 6308   CCcc 9555    - cmin 9880    shift cshi 13206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-ltxr 9698  df-sub 9882  df-shft 13207
This theorem is referenced by:  shftfn  13213
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