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Theorem shftfib 11583
Description: Value of a fiber of the relation  F. (Contributed by Mario Carneiro, 4-Nov-2013.)
Hypothesis
Ref Expression
shftfval.1  |-  F  e. 
_V
Assertion
Ref Expression
shftfib  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) " { B } )  =  ( F " { ( B  -  A ) } ) )

Proof of Theorem shftfib
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 shftfval.1 . . . . . . 7  |-  F  e. 
_V
21shftfval 11581 . . . . . 6  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
32breqd 4050 . . . . 5  |-  ( A  e.  CC  ->  ( B ( F  shift  A ) z  <->  B { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } z ) )
4 vex 2804 . . . . . 6  |-  z  e. 
_V
5 eleq1 2356 . . . . . . . 8  |-  ( x  =  B  ->  (
x  e.  CC  <->  B  e.  CC ) )
6 oveq1 5881 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  -  A )  =  ( B  -  A ) )
76breq1d 4049 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  -  A
) F y  <->  ( B  -  A ) F y ) )
85, 7anbi12d 691 . . . . . . 7  |-  ( x  =  B  ->  (
( x  e.  CC  /\  ( x  -  A
) F y )  <-> 
( B  e.  CC  /\  ( B  -  A
) F y ) ) )
9 breq2 4043 . . . . . . . 8  |-  ( y  =  z  ->  (
( B  -  A
) F y  <->  ( B  -  A ) F z ) )
109anbi2d 684 . . . . . . 7  |-  ( y  =  z  ->  (
( B  e.  CC  /\  ( B  -  A
) F y )  <-> 
( B  e.  CC  /\  ( B  -  A
) F z ) ) )
11 eqid 2296 . . . . . . 7  |-  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }
128, 10, 11brabg 4300 . . . . . 6  |-  ( ( B  e.  CC  /\  z  e.  _V )  ->  ( B { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } z  <->  ( B  e.  CC  /\  ( B  -  A ) F z ) ) )
134, 12mpan2 652 . . . . 5  |-  ( B  e.  CC  ->  ( B { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } z  <-> 
( B  e.  CC  /\  ( B  -  A
) F z ) ) )
143, 13sylan9bb 680 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ( F 
shift  A ) z  <->  ( B  e.  CC  /\  ( B  -  A ) F z ) ) )
15 ibar 490 . . . . 5  |-  ( B  e.  CC  ->  (
( B  -  A
) F z  <->  ( B  e.  CC  /\  ( B  -  A ) F z ) ) )
1615adantl 452 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( B  -  A ) F z  <-> 
( B  e.  CC  /\  ( B  -  A
) F z ) ) )
1714, 16bitr4d 247 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ( F 
shift  A ) z  <->  ( B  -  A ) F z ) )
1817abbidv 2410 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  { z  |  B
( F  shift  A ) z }  =  {
z  |  ( B  -  A ) F z } )
19 imasng 5051 . . 3  |-  ( B  e.  CC  ->  (
( F  shift  A )
" { B }
)  =  { z  |  B ( F 
shift  A ) z } )
2019adantl 452 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) " { B } )  =  {
z  |  B ( F  shift  A )
z } )
21 ovex 5899 . . 3  |-  ( B  -  A )  e. 
_V
22 imasng 5051 . . 3  |-  ( ( B  -  A )  e.  _V  ->  ( F " { ( B  -  A ) } )  =  { z  |  ( B  -  A ) F z } )
2321, 22mp1i 11 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( F " {
( B  -  A
) } )  =  { z  |  ( B  -  A ) F z } )
2418, 20, 233eqtr4d 2338 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) " { B } )  =  ( F " { ( B  -  A ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   {csn 3653   class class class wbr 4039   {copab 4092   "cima 4708  (class class class)co 5874   CCcc 8751    - cmin 9053    shift cshi 11577
This theorem is referenced by:  shftval  11585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-shft 11578
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