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Theorem grplactf1o 12905
Description: The left group action of element  A of group  G maps the underlying set  X of  G one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
grplact.1  |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a
) ) )
grplact.2  |-  X  =  ( Base `  G
)
grplact.3  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
grplactf1o  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( F `  A
) : X -1-1-onto-> X )
Distinct variable groups:    g, a, A    G, a, g    .+ , a,
g    X, a, g
Allowed substitution hints:    F( g, a)

Proof of Theorem grplactf1o
StepHypRef Expression
1 grplact.1 . . 3  |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a
) ) )
2 grplact.2 . . 3  |-  X  =  ( Base `  G
)
3 grplact.3 . . 3  |-  .+  =  ( +g  `  G )
4 eqid 2177 . . 3  |-  ( invg `  G )  =  ( invg `  G )
51, 2, 3, 4grplactcnv 12904 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( F `  A ) : X -1-1-onto-> X  /\  `' ( F `  A )  =  ( F `  ( ( invg `  G
) `  A )
) ) )
65simpld 112 1  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( F `  A
) : X -1-1-onto-> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    |-> cmpt 4063   `'ccnv 4624   -1-1-onto->wf1o 5214   ` cfv 5215  (class class class)co 5872   Basecbs 12454   +g cplusg 12528   Grpcgrp 12809   invgcminusg 12810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-plusg 12541  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813
This theorem is referenced by:  eqgen  13017
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