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Theorem shftfn 10258
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 4564 . . . . 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 5111 . . . . . 6  |-  ( F  Fn  B  ->  Fun  F )
43adantr 270 . . . . 5  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  Fun  F )
5 funmo 5030 . . . . . . 7  |-  ( Fun 
F  ->  E* w
( z  -  A
) F w )
6 vex 2622 . . . . . . . . . 10  |-  z  e. 
_V
7 vex 2622 . . . . . . . . . 10  |-  w  e. 
_V
8 eleq1 2150 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x  e.  CC  <->  z  e.  CC ) )
9 oveq1 5659 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
x  -  A )  =  ( z  -  A ) )
109breq1d 3855 . . . . . . . . . . 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 3849 . . . . . . . . . . 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 2088 . . . . . . . . . 10  |-  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }
156, 7, 11, 13, 14brab 4099 . . . . . . . . 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 2013 . . . . . . 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 1802 . . . . 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 5029 . . . 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 10255 . . . . 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 5037 . . 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 10256 . . 3  |-  ( A  e.  CC  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e. 
dom  F } )
29 fndm 5113 . . . . 5  |-  ( F  Fn  B  ->  dom  F  =  B )
3029eleq2d 2157 . . . 4  |-  ( F  Fn  B  ->  (
( x  -  A
)  e.  dom  F  <->  ( x  -  A )  e.  B ) )
3130rabbidv 2608 . . 3  |-  ( F  Fn  B  ->  { x  e.  CC  |  ( x  -  A )  e. 
dom  F }  =  { x  e.  CC  |  ( x  -  A )  e.  B } )
3228, 31sylan9eqr 2142 . 2  |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e.  B } )
33 df-fn 5018 . 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 1287    = wceq 1289    e. wcel 1438   E*wmo 1949   {crab 2363   _Vcvv 2619   class class class wbr 3845   {copab 3898   dom cdm 4438   Rel wrel 4443   Fun wfun 5009    Fn wfn 5010  (class class class)co 5652   CCcc 7348    - cmin 7653    shift cshi 10248
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-resscn 7437  ax-1cn 7438  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-sub 7655  df-shft 10249
This theorem is referenced by:  shftf  10264  seq3shft  10272
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