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Theorem grplactf1o 14664
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 2358 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
51, 2, 3, 4grplactcnv 14663 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( F `  A ) : X -1-1-onto-> X  /\  `' ( F `  A )  =  ( F `  ( ( inv g `  G
) `  A )
) ) )
65simpld 445 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 358    = wceq 1642    e. wcel 1710    e. cmpt 4158   `'ccnv 4770   -1-1-onto->wf1o 5336   ` cfv 5337  (class class class)co 5945   Basecbs 13245   +g cplusg 13305   Grpcgrp 14461   inv gcminusg 14462
This theorem is referenced by:  eqgen  14769  dchrsum2  20619  sumdchr2  20621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-riota 6391  df-0g 13503  df-mnd 14466  df-grp 14588  df-minusg 14589
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