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Theorem shftfibg 9649
Description: Value of a fiber of the relation  F. (Contributed by Jim Kingdon, 15-Aug-2021.)
Assertion
Ref Expression
shftfibg  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( F  shift  A )
" { B }
)  =  ( F
" { ( B  -  A ) } ) )

Proof of Theorem shftfibg
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 916 . . . . 5  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2 simp1 915 . . . . 5  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  F  e.  V )
3 simp3 917 . . . . 5  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
4 shftfvalg 9647 . . . . . . 7  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  ( F  shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
54breqd 3803 . . . . . 6  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  ( B ( F 
shift  A ) z  <->  B { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } z ) )
6 vex 2577 . . . . . . 7  |-  z  e. 
_V
7 eleq1 2116 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  e.  CC  <->  B  e.  CC ) )
8 oveq1 5547 . . . . . . . . . 10  |-  ( x  =  B  ->  (
x  -  A )  =  ( B  -  A ) )
98breq1d 3802 . . . . . . . . 9  |-  ( x  =  B  ->  (
( x  -  A
) F y  <->  ( B  -  A ) F y ) )
107, 9anbi12d 450 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  e.  CC  /\  ( x  -  A
) F y )  <-> 
( B  e.  CC  /\  ( B  -  A
) F y ) ) )
11 breq2 3796 . . . . . . . . 9  |-  ( y  =  z  ->  (
( B  -  A
) F y  <->  ( B  -  A ) F z ) )
1211anbi2d 445 . . . . . . . 8  |-  ( y  =  z  ->  (
( B  e.  CC  /\  ( B  -  A
) F y )  <-> 
( B  e.  CC  /\  ( B  -  A
) F z ) ) )
13 eqid 2056 . . . . . . . 8  |-  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }
1410, 12, 13brabg 4034 . . . . . . 7  |-  ( ( B  e.  CC  /\  z  e.  _V )  ->  ( B { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } z  <->  ( B  e.  CC  /\  ( B  -  A ) F z ) ) )
156, 14mpan2 409 . . . . . 6  |-  ( B  e.  CC  ->  ( B { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } z  <-> 
( B  e.  CC  /\  ( B  -  A
) F z ) ) )
165, 15sylan9bb 443 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  V )  /\  B  e.  CC )  ->  ( B ( F  shift  A )
z  <->  ( B  e.  CC  /\  ( B  -  A ) F z ) ) )
171, 2, 3, 16syl21anc 1145 . . . 4  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( B ( F  shift  A ) z  <->  ( B  e.  CC  /\  ( B  -  A ) F z ) ) )
18173anibar 1083 . . 3  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( B ( F  shift  A ) z  <->  ( B  -  A ) F z ) )
1918abbidv 2171 . 2  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  { z  |  B ( F 
shift  A ) z }  =  { z  |  ( B  -  A
) F z } )
20 imasng 4718 . . 3  |-  ( B  e.  CC  ->  (
( F  shift  A )
" { B }
)  =  { z  |  B ( F 
shift  A ) z } )
21203ad2ant3 938 . 2  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( F  shift  A )
" { B }
)  =  { z  |  B ( F 
shift  A ) z } )
223, 1subcld 7385 . . 3  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( B  -  A )  e.  CC )
23 imasng 4718 . . 3  |-  ( ( B  -  A )  e.  CC  ->  ( F " { ( B  -  A ) } )  =  { z  |  ( B  -  A ) F z } )
2422, 23syl 14 . 2  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( F " { ( B  -  A ) } )  =  { z  |  ( B  -  A ) F z } )
2519, 21, 243eqtr4d 2098 1  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( F  shift  A )
" { B }
)  =  ( F
" { ( B  -  A ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   {cab 2042   _Vcvv 2574   {csn 3403   class class class wbr 3792   {copab 3845   "cima 4376  (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-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: (None)
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