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Theorem trdom2 24791
Description: The domain of a right translation. The term  A is a constant:  x is not present. (Contributed by FL, 21-Jun-2010.)
Hypotheses
Ref Expression
trfun.2  |-  F  =  ( x  e.  X  |->  ( x G A ) )
trinv.1  |-  X  =  ran  G
Assertion
Ref Expression
trdom2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  dom  F  =  X )
Distinct variable groups:    x, A    x, G    x, X
Allowed substitution hint:    F( x)

Proof of Theorem trdom2
StepHypRef Expression
1 ovex 5845 . . . 4  |-  ( x G A )  e. 
_V
21a1i 12 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( x G A )  e.  _V )
32ralrimiva 2628 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  A. x  e.  X  ( x G A )  e.  _V )
4 trfun.2 . . 3  |-  F  =  ( x  e.  X  |->  ( x G A ) )
54cmpdom 24543 . 2  |-  ( A. x  e.  X  (
x G A )  e.  _V  <->  dom  F  =  X )
63, 5sylib 190 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  dom  F  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   A.wral 2545   _Vcvv 2790    e. cmpt 4079   dom cdm 4689   ran crn 4690  (class class class)co 5820   GrpOpcgr 20846
This theorem is referenced by:  imtr  24798
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5224  df-fn 5225  df-fv 5230  df-ov 5823
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