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Theorem rngapm 25473
Description: The range of the action of a particular group element equals the range of the action. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 3-May-2015.)
Hypothesis
Ref Expression
gapm2.1  |-  X  =  ( Base `  G
)
Assertion
Ref Expression
rngapm  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  ->  ran  ( ( cur1 `  .(+)  ) `  A )  =  ran  .(+) 
)

Proof of Theorem rngapm
StepHypRef Expression
1 gapm2.1 . . . 4  |-  X  =  ( Base `  G
)
21gapm2 25472 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  -> 
( ( cur1 `  .(+)  ) `  A ) : Y -1-1-onto-> Y
)
3 f1ofo 5495 . . 3  |-  ( ( ( cur1 `  .(+)  ) `  A ) : Y -1-1-onto-> Y  ->  ( ( cur1 `  .(+)  ) `  A ) : Y -onto-> Y )
4 forn 5470 . . 3  |-  ( ( ( cur1 `  .(+)  ) `  A ) : Y -onto-> Y  ->  ran  ( ( cur1 `  .(+)  ) `  A )  =  Y )
52, 3, 43syl 18 . 2  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  ->  ran  ( ( cur1 `  .(+)  ) `  A )  =  Y )
61gafo 14766 . . . 4  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) -onto-> Y )
76adantr 451 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  ->  .(+)  : ( X  X.  Y ) -onto-> Y )
8 forn 5470 . . 3  |-  (  .(+)  : ( X  X.  Y
) -onto-> Y  ->  ran  .(+)  =  Y )
97, 8syl 15 . 2  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  ->  ran  .(+)  =  Y )
105, 9eqtr4d 2331 1  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  ->  ran  ( ( cur1 `  .(+)  ) `  A )  =  ran  .(+) 
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468    X. cxp 4703   ran crn 4706   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   Basecbs 13164    GrpAct cga 14759   cur1ccur1 25297
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-rep 4147  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-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-map 6790  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-ga 14760  df-cur1 25299
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