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Theorem gafo 14750
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 14749 . 2  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) --> Y )
3 gagrp 14746 . . . . . 6  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  G  e.  Grp )
43adantr 451 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  G  e.  Grp )
5 eqid 2283 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
61, 5grpidcl 14510 . . . . 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 14748 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  (
( 0g `  G
)  .(+)  x )  =  x )
109eqcomd 2288 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  x  =  ( ( 0g
`  G )  .(+)  x ) )
11 rspceov 5893 . . . 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 2626 . 2  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  A. x  e.  Y  E. y  e.  X  E. z  e.  Y  x  =  ( y  .(+)  z ) )
14 foov 5994 . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    X. cxp 4687   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   Basecbs 13148   0gc0g 13400   Grpcgrp 14362    GrpAct cga 14743
This theorem is referenced by:  rngapm  24782
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  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-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-map 6774  df-0g 13404  df-mnd 14367  df-grp 14489  df-ga 14744
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