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Theorem trdom2 24557
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 5735 . . . 4  |-  ( x G A )  e. 
_V
21a1i 12 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( x G A )  e.  _V )
32ralrimiva 2588 . 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 24309 . 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 2509   _Vcvv 2727    e. cmpt 3974   dom cdm 4580   ran crn 4581  (class class class)co 5710   GrpOpcgr 20683
This theorem is referenced by:  imtr  24564
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-fun 4602  df-fn 4603  df-fv 4608  df-ov 5713
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