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Theorem gafo 14766
Description: A group action is onto its base set. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypothesis
Ref Expression
gaf.1  |-  X  =  ( Base `  G
)
Assertion
Ref Expression
gafo  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) -onto-> Y )

Proof of Theorem gafo
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gaf.1 . . 3  |-  X  =  ( Base `  G
)
21gaf 14765 . 2  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) --> Y )
3 gagrp 14762 . . . . . 6  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  G  e.  Grp )
43adantr 451 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  G  e.  Grp )
5 eqid 2296 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
61, 5grpidcl 14526 . . . . 5  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  X )
74, 6syl 15 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  ( 0g `  G )  e.  X )
8 simpr 447 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  x  e.  Y )
95gagrpid 14764 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  (
( 0g `  G
)  .(+)  x )  =  x )
109eqcomd 2301 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  x  =  ( ( 0g
`  G )  .(+)  x ) )
11 rspceov 5909 . . . 4  |-  ( ( ( 0g `  G
)  e.  X  /\  x  e.  Y  /\  x  =  ( ( 0g `  G )  .(+)  x ) )  ->  E. y  e.  X  E. z  e.  Y  x  =  ( y  .(+)  z ) )
127, 8, 10, 11syl3anc 1182 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  E. y  e.  X  E. z  e.  Y  x  =  ( y  .(+)  z ) )
1312ralrimiva 2639 . 2  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  A. x  e.  Y  E. y  e.  X  E. z  e.  Y  x  =  ( y  .(+)  z ) )
14 foov 6010 . 2  |-  (  .(+)  : ( X  X.  Y
) -onto-> Y  <->  (  .(+)  : ( X  X.  Y ) --> Y  /\  A. x  e.  Y  E. y  e.  X  E. z  e.  Y  x  =  ( y  .(+)  z ) ) )
152, 13, 14sylanbrc 645 1  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) -onto-> Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    X. cxp 4703   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   Basecbs 13164   0gc0g 13416   Grpcgrp 14378    GrpAct cga 14759
This theorem is referenced by:  rngapm  25473
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-reu 2563  df-rmo 2564  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-iun 3923  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-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-map 6790  df-0g 13420  df-mnd 14383  df-grp 14505  df-ga 14760
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