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Theorem fnmgp 15652
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 6108 . 2  |-  ( x sSet  <. ( +g  `  ndx ) ,  ( .r `  x ) >. )  e.  _V
2 df-mgp 15651 . 2  |- mulGrp  =  ( x  e.  _V  |->  ( x sSet  <. ( +g  `  ndx ) ,  ( .r `  x ) >. )
)
31, 2fnmpti 5575 1  |- mulGrp  Fn  _V
Colors of variables: wff set class
Syntax hints:   _Vcvv 2958   <.cop 3819    Fn wfn 5451   ` cfv 5456  (class class class)co 6083   ndxcnx 13468   sSet csts 13469   +g cplusg 13531   .rcmulr 13532  mulGrpcmgp 15650
This theorem is referenced by:  rngidval  15668  mgpf  15677  prdsmgp  15718  prdscrngd  15721  pws1  15724  pwsmgp  15726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464  df-ov 6086  df-mgp 15651
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