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Theorem imtr 25398
Description: The image of a set through a translation. (Contributed by FL, 30-Dec-2010.)
Hypotheses
Ref Expression
trfun.2  |-  F  =  ( x  e.  X  |->  ( x G A ) )
trinv.1  |-  X  =  ran  G
Assertion
Ref Expression
imtr  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( F " B
)  =  { a  |  E. u  e.  B  a  =  ( F `  u ) } )
Distinct variable groups:    x, A    B, a, u    F, a, u    x, G    x, X
Allowed substitution hints:    A( u, a)    B( x)    F( x)    G( u, a)    X( u, a)

Proof of Theorem imtr
StepHypRef Expression
1 trfun.2 . . . 4  |-  F  =  ( x  e.  X  |->  ( x G A ) )
21funmpt2 5291 . . 3  |-  Fun  F
3 trinv.1 . . . . . 6  |-  X  =  ran  G
41, 3trdom2 25391 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  dom  F  =  X )
5 pweq 3628 . . . . . . . . 9  |-  ( X  =  dom  F  ->  ~P X  =  ~P dom  F )
65eleq2d 2350 . . . . . . . 8  |-  ( X  =  dom  F  -> 
( B  e.  ~P X 
<->  B  e.  ~P dom  F ) )
76biimpd 198 . . . . . . 7  |-  ( X  =  dom  F  -> 
( B  e.  ~P X  ->  B  e.  ~P dom  F ) )
87eqcoms 2286 . . . . . 6  |-  ( dom 
F  =  X  -> 
( B  e.  ~P X  ->  B  e.  ~P dom  F ) )
9 elpwi 3633 . . . . . 6  |-  ( B  e.  ~P dom  F  ->  B  C_  dom  F )
108, 9syl6 29 . . . . 5  |-  ( dom 
F  =  X  -> 
( B  e.  ~P X  ->  B  C_  dom  F ) )
114, 10syl 15 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( B  e.  ~P X  ->  B  C_  dom  F ) )
12113impia 1148 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  ->  B  C_  dom  F )
13 dfimafn 5571 . . 3  |-  ( ( Fun  F  /\  B  C_ 
dom  F )  -> 
( F " B
)  =  { a  |  E. u  e.  B  ( F `  u )  =  a } )
142, 12, 13sylancr 644 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( F " B
)  =  { a  |  E. u  e.  B  ( F `  u )  =  a } )
15 eqcom 2285 . . . 4  |-  ( ( F `  u )  =  a  <->  a  =  ( F `  u ) )
1615rexbii 2568 . . 3  |-  ( E. u  e.  B  ( F `  u )  =  a  <->  E. u  e.  B  a  =  ( F `  u ) )
1716abbii 2395 . 2  |-  { a  |  E. u  e.  B  ( F `  u )  =  a }  =  { a  |  E. u  e.  B  a  =  ( F `  u ) }
1814, 17syl6eq 2331 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( F " B
)  =  { a  |  E. u  e.  B  a  =  ( F `  u ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    C_ wss 3152   ~Pcpw 3625    e. cmpt 4077   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861
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