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Theorem trdom2 24744
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 5803 . . . 4  |-  ( x G A )  e. 
_V
21a1i 12 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( x G A )  e.  _V )
32ralrimiva 2599 . 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 24496 . 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 1619    e. wcel 1621   A.wral 2516   _Vcvv 2757    e. cmpt 4037   dom cdm 4647   ran crn 4648  (class class class)co 5778   GrpOpcgr 20799
This theorem is referenced by:  imtr  24751
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-fun 4669  df-fn 4670  df-fv 4675  df-ov 5781
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