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Theorem ovshftex 9648
Description: Existence of the result of applying shift. (Contributed by Jim Kingdon, 15-Aug-2021.)
Assertion
Ref Expression
ovshftex  |-  ( ( F  e.  V  /\  A  e.  CC )  ->  ( F  shift  A )  e.  _V )

Proof of Theorem ovshftex
Dummy variables  u  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 shftfvalg 9647 . . 3  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  ( F  shift  A )  =  { <. z ,  w >.  |  (
z  e.  CC  /\  ( z  -  A
) F w ) } )
21ancoms 259 . 2  |-  ( ( F  e.  V  /\  A  e.  CC )  ->  ( F  shift  A )  =  { <. z ,  w >.  |  (
z  e.  CC  /\  ( z  -  A
) F w ) } )
3 cnex 7063 . . . 4  |-  CC  e.  _V
43a1i 9 . . 3  |-  ( ( F  e.  V  /\  A  e.  CC )  ->  CC  e.  _V )
5 rnexg 4625 . . . . 5  |-  ( F  e.  V  ->  ran  F  e.  _V )
65ad2antrr 465 . . . 4  |-  ( ( ( F  e.  V  /\  A  e.  CC )  /\  z  e.  CC )  ->  ran  F  e.  _V )
7 vex 2577 . . . . . . . 8  |-  u  e. 
_V
8 breq2 3796 . . . . . . . 8  |-  ( w  =  u  ->  (
( z  -  A
) F w  <->  ( z  -  A ) F u ) )
97, 8elab 2710 . . . . . . 7  |-  ( u  e.  { w  |  ( z  -  A
) F w }  <->  ( z  -  A ) F u )
10 simpr 107 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  z  e.  CC )
11 simpl 106 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  A  e.  CC )
1210, 11subcld 7385 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( z  -  A
)  e.  CC )
13 brelrng 4593 . . . . . . . . . 10  |-  ( ( ( z  -  A
)  e.  CC  /\  u  e.  _V  /\  (
z  -  A ) F u )  ->  u  e.  ran  F )
147, 13mp3an2 1231 . . . . . . . . 9  |-  ( ( ( z  -  A
)  e.  CC  /\  ( z  -  A
) F u )  ->  u  e.  ran  F )
1512, 14sylan 271 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  z  e.  CC )  /\  ( z  -  A ) F u )  ->  u  e.  ran  F )
1615ex 112 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( z  -  A ) F u  ->  u  e.  ran  F ) )
179, 16syl5bi 145 . . . . . 6  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( u  e.  {
w  |  ( z  -  A ) F w }  ->  u  e.  ran  F ) )
1817ssrdv 2979 . . . . 5  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  { w  |  ( z  -  A ) F w }  C_  ran  F )
1918adantll 453 . . . 4  |-  ( ( ( F  e.  V  /\  A  e.  CC )  /\  z  e.  CC )  ->  { w  |  ( z  -  A
) F w }  C_ 
ran  F )
206, 19ssexd 3925 . . 3  |-  ( ( ( F  e.  V  /\  A  e.  CC )  /\  z  e.  CC )  ->  { w  |  ( z  -  A
) F w }  e.  _V )
214, 20opabex3d 5776 . 2  |-  ( ( F  e.  V  /\  A  e.  CC )  ->  { <. z ,  w >.  |  ( z  e.  CC  /\  ( z  -  A ) F w ) }  e.  _V )
222, 21eqeltrd 2130 1  |-  ( ( F  e.  V  /\  A  e.  CC )  ->  ( F  shift  A )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   {cab 2042   _Vcvv 2574    C_ wss 2945   class class class wbr 3792   {copab 3845   ran crn 4374  (class class class)co 5540   CCcc 6945    - cmin 7245    shift cshi 9643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-sub 7247  df-shft 9644
This theorem is referenced by:  2shfti  9660  climshftlemg  10054  climshft  10056  climshft2  10058
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