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Theorem shftfib 10723
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 10721 . . . . . 6  |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } )
32breqd 3976 . . . . 5  |-  ( A  e.  CC  ->  ( B ( F  shift  A ) z  <->  B { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } z ) )
4 vex 2715 . . . . . 6  |-  z  e. 
_V
5 eleq1 2220 . . . . . . . 8  |-  ( x  =  B  ->  (
x  e.  CC  <->  B  e.  CC ) )
6 oveq1 5831 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  -  A )  =  ( B  -  A ) )
76breq1d 3975 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  -  A
) F y  <->  ( B  -  A ) F y ) )
85, 7anbi12d 465 . . . . . . 7  |-  ( x  =  B  ->  (
( x  e.  CC  /\  ( x  -  A
) F y )  <-> 
( B  e.  CC  /\  ( B  -  A
) F y ) ) )
9 breq2 3969 . . . . . . . 8  |-  ( y  =  z  ->  (
( B  -  A
) F y  <->  ( B  -  A ) F z ) )
109anbi2d 460 . . . . . . 7  |-  ( y  =  z  ->  (
( B  e.  CC  /\  ( B  -  A
) F y )  <-> 
( B  e.  CC  /\  ( B  -  A
) F z ) ) )
11 eqid 2157 . . . . . . 7  |-  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }
128, 10, 11brabg 4229 . . . . . 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 422 . . . . 5  |-  ( B  e.  CC  ->  ( B { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } z  <-> 
( B  e.  CC  /\  ( B  -  A
) F z ) ) )
143, 13sylan9bb 458 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ( F 
shift  A ) z  <->  ( B  e.  CC  /\  ( B  -  A ) F z ) ) )
15 ibar 299 . . . . 5  |-  ( B  e.  CC  ->  (
( B  -  A
) F z  <->  ( B  e.  CC  /\  ( B  -  A ) F z ) ) )
1615adantl 275 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( B  -  A ) F z  <-> 
( B  e.  CC  /\  ( B  -  A
) F z ) ) )
1714, 16bitr4d 190 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ( F 
shift  A ) z  <->  ( B  -  A ) F z ) )
1817abbidv 2275 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  { z  |  B
( F  shift  A ) z }  =  {
z  |  ( B  -  A ) F z } )
19 imasng 4951 . . 3  |-  ( B  e.  CC  ->  (
( F  shift  A )
" { B }
)  =  { z  |  B ( F 
shift  A ) z } )
2019adantl 275 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) " { B } )  =  {
z  |  B ( F  shift  A )
z } )
21 simpr 109 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
22 simpl 108 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2321, 22subcld 8186 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  -  A
)  e.  CC )
24 imasng 4951 . . 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 2200 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 103    <-> wb 104    = wceq 1335    e. wcel 2128   {cab 2143   _Vcvv 2712   {csn 3560   class class class wbr 3965   {copab 4024   "cima 4589  (class class class)co 5824   CCcc 7730    - cmin 8046    shift cshi 10714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-resscn 7824  ax-1cn 7825  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-sub 8048  df-shft 10715
This theorem is referenced by:  shftval  10725
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