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Theorem shftfib 10834
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 10832 . . . . . 6  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
32breqd 4016 . . . . 5  |-  ( A  e.  CC  ->  ( B ( F  shift  A ) z  <->  B { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } z ) )
4 vex 2742 . . . . . 6  |-  z  e. 
_V
5 eleq1 2240 . . . . . . . 8  |-  ( x  =  B  ->  (
x  e.  CC  <->  B  e.  CC ) )
6 oveq1 5884 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  -  A )  =  ( B  -  A ) )
76breq1d 4015 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  -  A
) F y  <->  ( B  -  A ) F y ) )
85, 7anbi12d 473 . . . . . . 7  |-  ( x  =  B  ->  (
( x  e.  CC  /\  ( x  -  A
) F y )  <-> 
( B  e.  CC  /\  ( B  -  A
) F y ) ) )
9 breq2 4009 . . . . . . . 8  |-  ( y  =  z  ->  (
( B  -  A
) F y  <->  ( B  -  A ) F z ) )
109anbi2d 464 . . . . . . 7  |-  ( y  =  z  ->  (
( B  e.  CC  /\  ( B  -  A
) F y )  <-> 
( B  e.  CC  /\  ( B  -  A
) F z ) ) )
11 eqid 2177 . . . . . . 7  |-  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }
128, 10, 11brabg 4271 . . . . . 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 425 . . . . 5  |-  ( B  e.  CC  ->  ( B { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } z  <-> 
( B  e.  CC  /\  ( B  -  A
) F z ) ) )
143, 13sylan9bb 462 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ( F 
shift  A ) z  <->  ( B  e.  CC  /\  ( B  -  A ) F z ) ) )
15 ibar 301 . . . . 5  |-  ( B  e.  CC  ->  (
( B  -  A
) F z  <->  ( B  e.  CC  /\  ( B  -  A ) F z ) ) )
1615adantl 277 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( B  -  A ) F z  <-> 
( B  e.  CC  /\  ( B  -  A
) F z ) ) )
1714, 16bitr4d 191 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ( F 
shift  A ) z  <->  ( B  -  A ) F z ) )
1817abbidv 2295 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  { z  |  B
( F  shift  A ) z }  =  {
z  |  ( B  -  A ) F z } )
19 imasng 4995 . . 3  |-  ( B  e.  CC  ->  (
( F  shift  A )
" { B }
)  =  { z  |  B ( F 
shift  A ) z } )
2019adantl 277 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) " { B } )  =  {
z  |  B ( F  shift  A )
z } )
21 simpr 110 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
22 simpl 109 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2321, 22subcld 8270 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  -  A
)  e.  CC )
24 imasng 4995 . . 3  |-  ( ( B  -  A )  e.  CC  ->  ( F " { ( B  -  A ) } )  =  { z  |  ( B  -  A ) F z } )
2523, 24syl 14 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( F " {
( B  -  A
) } )  =  { z  |  ( B  -  A ) F z } )
2618, 20, 253eqtr4d 2220 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) " { B } )  =  ( F " { ( B  -  A ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   {cab 2163   _Vcvv 2739   {csn 3594   class class class wbr 4005   {copab 4065   "cima 4631  (class class class)co 5877   CCcc 7811    - cmin 8130    shift cshi 10825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-resscn 7905  ax-1cn 7906  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-sub 8132  df-shft 10826
This theorem is referenced by:  shftval  10836
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