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Theorem gafo 15928
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 15927 . 2  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) --> Y )
3 gagrp 15924 . . . . . 6  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  G  e.  Grp )
43adantr 465 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  G  e.  Grp )
5 eqid 2452 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
61, 5grpidcl 15680 . . . . 5  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  X )
74, 6syl 16 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  ( 0g `  G )  e.  X )
8 simpr 461 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  x  e.  Y )
95gagrpid 15926 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  (
( 0g `  G
)  .(+)  x )  =  x )
109eqcomd 2460 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  x  =  ( ( 0g
`  G )  .(+)  x ) )
11 rspceov 6232 . . . 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 1219 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  x  e.  Y )  ->  E. y  e.  X  E. z  e.  Y  x  =  ( y  .(+)  z ) )
1312ralrimiva 2827 . 2  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  A. x  e.  Y  E. y  e.  X  E. z  e.  Y  x  =  ( y  .(+)  z ) )
14 foov 6342 . 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 664 1  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) -onto-> Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2796   E.wrex 2797    X. cxp 4941   -->wf 5517   -onto->wfo 5519   ` cfv 5521  (class class class)co 6195   Basecbs 14287   0gc0g 14492   Grpcgrp 15524    GrpAct cga 15921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-fo 5527  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-map 7321  df-0g 14494  df-mnd 15529  df-grp 15659  df-ga 15922
This theorem is referenced by: (None)
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