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Theorem trdom2 25494
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 5899 . . . 4  |-  ( x G A )  e. 
_V
21a1i 10 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( x G A )  e.  _V )
32ralrimiva 2639 . 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 25246 . 2  |-  ( A. x  e.  X  (
x G A )  e.  _V  <->  dom  F  =  X )
63, 5sylib 188 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  dom  F  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    e. cmpt 4093   dom cdm 4705   ran crn 4706  (class class class)co 5874   GrpOpcgr 20869
This theorem is referenced by:  imtr  25501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5877
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