ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  shftfibg Unicode version

Theorem shftfibg 10560
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 967 . . . . 5  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2 simp1 966 . . . . 5  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  F  e.  V )
3 simp3 968 . . . . 5  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
4 shftfvalg 10558 . . . . . . 7  |-  ( ( A  e.  CC  /\  F  e.  V )  ->  ( F  shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) } )
54breqd 3910 . . . . . 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 2663 . . . . . . 7  |-  z  e. 
_V
7 eleq1 2180 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  e.  CC  <->  B  e.  CC ) )
8 oveq1 5749 . . . . . . . . . 10  |-  ( x  =  B  ->  (
x  -  A )  =  ( B  -  A ) )
98breq1d 3909 . . . . . . . . 9  |-  ( x  =  B  ->  (
( x  -  A
) F y  <->  ( B  -  A ) F y ) )
107, 9anbi12d 464 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  e.  CC  /\  ( x  -  A
) F y )  <-> 
( B  e.  CC  /\  ( B  -  A
) F y ) ) )
11 breq2 3903 . . . . . . . . 9  |-  ( y  =  z  ->  (
( B  -  A
) F y  <->  ( B  -  A ) F z ) )
1211anbi2d 459 . . . . . . . 8  |-  ( y  =  z  ->  (
( B  e.  CC  /\  ( B  -  A
) F y )  <-> 
( B  e.  CC  /\  ( B  -  A
) F z ) ) )
13 eqid 2117 . . . . . . . 8  |-  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A
) F y ) }
1410, 12, 13brabg 4161 . . . . . . 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 421 . . . . . 6  |-  ( B  e.  CC  ->  ( B { <. x ,  y
>.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) } z  <-> 
( B  e.  CC  /\  ( B  -  A
) F z ) ) )
165, 15sylan9bb 457 . . . . 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 1200 . . . 4  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( B ( F  shift  A ) z  <->  ( B  e.  CC  /\  ( B  -  A ) F z ) ) )
18173anibar 1134 . . 3  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( B ( F  shift  A ) z  <->  ( B  -  A ) F z ) )
1918abbidv 2235 . 2  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  { z  |  B ( F 
shift  A ) z }  =  { z  |  ( B  -  A
) F z } )
20 imasng 4874 . . 3  |-  ( B  e.  CC  ->  (
( F  shift  A )
" { B }
)  =  { z  |  B ( F 
shift  A ) z } )
21203ad2ant3 989 . 2  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( F  shift  A )
" { B }
)  =  { z  |  B ( F 
shift  A ) z } )
223, 1subcld 8041 . . 3  |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( B  -  A )  e.  CC )
23 imasng 4874 . . 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 2160 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 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   {cab 2103   _Vcvv 2660   {csn 3497   class class class wbr 3899   {copab 3958   "cima 4512  (class class class)co 5742   CCcc 7586    - cmin 7901    shift cshi 10554
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-resscn 7680  ax-1cn 7681  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-sub 7903  df-shft 10555
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator