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Theorem shftfn 9850
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 4492 . . . . 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 5027 . . . . . 6  |-  ( F  Fn  B  ->  Fun  F )
43adantr 270 . . . . 5  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Fun  F )
5 funmo 4947 . . . . . . 7  |-  ( Fun 
F  ->  E* w
( z  -  A
) F w )
6 vex 2605 . . . . . . . . . 10  |-  z  e. 
_V
7 vex 2605 . . . . . . . . . 10  |-  w  e. 
_V
8 eleq1 2142 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x  e.  CC  <->  z  e.  CC ) )
9 oveq1 5550 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
x  -  A )  =  ( z  -  A ) )
109breq1d 3803 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( x  -  A
) F y  <->  ( z  -  A ) F y ) )
118, 10anbi12d 457 . . . . . . . . . 10  |-  ( x  =  z  ->  (
( x  e.  CC  /\  ( x  -  A
) F y )  <-> 
( z  e.  CC  /\  ( z  -  A
) F y ) ) )
12 breq2 3797 . . . . . . . . . . 11  |-  ( y  =  w  ->  (
( z  -  A
) F y  <->  ( z  -  A ) F w ) )
1312anbi2d 452 . . . . . . . . . 10  |-  ( y  =  w  ->  (
( z  e.  CC  /\  ( z  -  A
) F y )  <-> 
( z  e.  CC  /\  ( z  -  A
) F w ) ) )
14 eqid 2082 . . . . . . . . . 10  |-  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }
156, 7, 11, 13, 14brab 4035 . . . . . . . . 9  |-  ( z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w  <->  ( z  e.  CC  /\  ( z  -  A
) F w ) )
1615simprbi 269 . . . . . . . 8  |-  ( z { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } w  ->  ( z  -  A
) F w )
1716moimi 2007 . . . . . . 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 1796 . . . . 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 4946 . . . 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 408 . . 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 9847 . . . . 5  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
2524adantl 271 . . . 4  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( F  shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
2625funeqd 4953 . . 3  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( Fun  ( F 
shift  A )  <->  Fun  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } ) )
2722, 26mpbird 165 . 2  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Fun  ( F  shift  A ) )
2823shftdm 9848 . . 3  |-  ( A  e.  CC  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e. 
dom  F } )
29 fndm 5029 . . . . 5  |-  ( F  Fn  B  ->  dom  F  =  B )
3029eleq2d 2149 . . . 4  |-  ( F  Fn  B  ->  (
( x  -  A
)  e.  dom  F  <->  ( x  -  A )  e.  B ) )
3130rabbidv 2594 . . 3  |-  ( F  Fn  B  ->  { x  e.  CC  |  ( x  -  A )  e. 
dom  F }  =  { x  e.  CC  |  ( x  -  A )  e.  B } )
3228, 31sylan9eqr 2136 . 2  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e.  B } )
33 df-fn 4935 . 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 408 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 102   A.wal 1283    = wceq 1285    e. wcel 1434   E*wmo 1943   {crab 2353   _Vcvv 2602   class class class wbr 3793   {copab 3846   dom cdm 4371   Rel wrel 4376   Fun wfun 4926    Fn wfn 4927  (class class class)co 5543   CCcc 7041    - cmin 7346    shift cshi 9840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-resscn 7130  ax-1cn 7131  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-sub 7348  df-shft 9841
This theorem is referenced by:  shftf  9856
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