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Theorem shftfn 9425
Description: Functionality and domain of a sequence shifted by  A. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
Hypothesis
Ref Expression
shftfval.1  |-  F  e. 
_V
Assertion
Ref Expression
shftfn  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( F  shift  A )  Fn  { x  e.  CC  |  ( x  -  A )  e.  B } )
Distinct variable groups:    x, A    x, F    x, B

Proof of Theorem shftfn
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 4464 . . . . 5  |-  Rel  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }
21a1i 9 . . . 4  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Rel  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
3 fnfun 4996 . . . . . 6  |-  ( F  Fn  B  ->  Fun  F )
43adantr 261 . . . . 5  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Fun  F )
5 funmo 4917 . . . . . . 7  |-  ( Fun 
F  ->  E* w
( z  -  A
) F w )
6 vex 2560 . . . . . . . . . 10  |-  z  e. 
_V
7 vex 2560 . . . . . . . . . 10  |-  w  e. 
_V
8 eleq1 2100 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x  e.  CC  <->  z  e.  CC ) )
9 oveq1 5519 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
x  -  A )  =  ( z  -  A ) )
109breq1d 3774 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( x  -  A
) F y  <->  ( z  -  A ) F y ) )
118, 10anbi12d 442 . . . . . . . . . 10  |-  ( x  =  z  ->  (
( x  e.  CC  /\  ( x  -  A
) F y )  <-> 
( z  e.  CC  /\  ( z  -  A
) F y ) ) )
12 breq2 3768 . . . . . . . . . . 11  |-  ( y  =  w  ->  (
( z  -  A
) F y  <->  ( z  -  A ) F w ) )
1312anbi2d 437 . . . . . . . . . 10  |-  ( y  =  w  ->  (
( z  e.  CC  /\  ( z  -  A
) F y )  <-> 
( z  e.  CC  /\  ( z  -  A
) F w ) ) )
14 eqid 2040 . . . . . . . . . 10  |-  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }
156, 7, 11, 13, 14brab 4009 . . . . . . . . 9  |-  ( z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w  <->  ( z  e.  CC  /\  ( z  -  A
) F w ) )
1615simprbi 260 . . . . . . . 8  |-  ( z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w  ->  ( z  -  A
) F w )
1716moimi 1965 . . . . . . 7  |-  ( E* w ( z  -  A ) F w  ->  E* w  z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
)
185, 17syl 14 . . . . . 6  |-  ( Fun 
F  ->  E* w  z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
)
1918alrimiv 1754 . . . . 5  |-  ( Fun 
F  ->  A. z E* w  z { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
)
204, 19syl 14 . . . 4  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  A. z E* w  z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
)
21 dffun6 4916 . . . 4  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  <->  ( Rel  {
<. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }  /\  A. z E* w  z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w
) )
222, 20, 21sylanbrc 394 . . 3  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Fun  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
23 shftfval.1 . . . . . 6  |-  F  e. 
_V
2423shftfval 9422 . . . . 5  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
2524adantl 262 . . . 4  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( F  shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
2625funeqd 4923 . . 3  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( Fun  ( F 
shift  A )  <->  Fun  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } ) )
2722, 26mpbird 156 . 2  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Fun  ( F  shift  A ) )
2823shftdm 9423 . . 3  |-  ( A  e.  CC  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e. 
dom  F } )
29 fndm 4998 . . . . 5  |-  ( F  Fn  B  ->  dom  F  =  B )
3029eleq2d 2107 . . . 4  |-  ( F  Fn  B  ->  (
( x  -  A
)  e.  dom  F  <->  ( x  -  A )  e.  B ) )
3130rabbidv 2549 . . 3  |-  ( F  Fn  B  ->  { x  e.  CC  |  ( x  -  A )  e. 
dom  F }  =  { x  e.  CC  |  ( x  -  A )  e.  B } )
3228, 31sylan9eqr 2094 . 2  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e.  B } )
33 df-fn 4905 . 2  |-  ( ( F  shift  A )  Fn  { x  e.  CC  |  ( x  -  A )  e.  B } 
<->  ( Fun  ( F 
shift  A )  /\  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e.  B } ) )
3427, 32, 33sylanbrc 394 1  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( F  shift  A )  Fn  { x  e.  CC  |  ( x  -  A )  e.  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97   A.wal 1241    = wceq 1243    e. wcel 1393   E*wmo 1901   {crab 2310   _Vcvv 2557   class class class wbr 3764   {copab 3817   dom cdm 4345   Rel wrel 4350   Fun wfun 4896    Fn wfn 4897  (class class class)co 5512   CCcc 6887    - cmin 7182    shift cshi 9415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-sub 7184  df-shft 9416
This theorem is referenced by:  shftf  9431
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