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Theorem gaset 14763
Description: The right argument of a group action is a set. (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
gaset  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  Y  e.  _V )

Proof of Theorem gaset
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2296 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2296 . . . 4  |-  ( 0g
`  G )  =  ( 0g `  G
)
41, 2, 3isga 14761 . . 3  |-  (  .(+)  e.  ( G  GrpAct  Y )  <-> 
( ( G  e. 
Grp  /\  Y  e.  _V )  /\  (  .(+)  : ( ( Base `  G )  X.  Y
) --> Y  /\  A. x  e.  Y  (
( ( 0g `  G )  .(+)  x )  =  x  /\  A. y  e.  ( Base `  G ) A. z  e.  ( Base `  G
) ( ( y ( +g  `  G
) z )  .(+)  x )  =  ( y 
.(+)  ( z  .(+)  x ) ) ) ) ) )
54simplbi 446 . 2  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  ( G  e. 
Grp  /\  Y  e.  _V ) )
65simprd 449 1  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  Y  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378    GrpAct cga 14759
This theorem is referenced by:  gass  14771  gasubg  14772  galactghm  14799  gapm2  25472  curgrpact  25475
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-ga 14760
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