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Theorem fnmgp 17015
Description: The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
Assertion
Ref Expression
fnmgp  |- mulGrp  Fn  _V

Proof of Theorem fnmgp
StepHypRef Expression
1 ovex 6320 . 2  |-  ( x sSet  <. ( +g  `  ndx ) ,  ( .r `  x ) >. )  e.  _V
2 df-mgp 17014 . 2  |- mulGrp  =  ( x  e.  _V  |->  ( x sSet  <. ( +g  `  ndx ) ,  ( .r `  x ) >. )
)
31, 2fnmpti 5715 1  |- mulGrp  Fn  _V
Colors of variables: wff setvar class
Syntax hints:   _Vcvv 3118   <.cop 4039    Fn wfn 5589   ` cfv 5594  (class class class)co 6295   ndxcnx 14504   sSet csts 14505   +g cplusg 14572   .rcmulr 14573  mulGrpcmgp 17013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-ov 6298  df-mgp 17014
This theorem is referenced by:  rngidval  17027  mgpf  17082  prdsmgp  17131  prdscrngd  17134  pws1  17137  pwsmgp  17139
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