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Theorem trdom2 25391
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 5883 . . . 4  |-  ( x G A )  e. 
_V
21a1i 10 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( x G A )  e.  _V )
32ralrimiva 2626 . 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 25143 . 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    e. cmpt 4077   dom cdm 4689   ran crn 4690  (class class class)co 5858   GrpOpcgr 20853
This theorem is referenced by:  imtr  25398
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-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-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861
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