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Theorem grpinvf1o 14538
Description: The group inverse is a one-to-one onto function. (Contributed by NM, 22-Oct-2014.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
grpinvinv.b  |-  B  =  ( Base `  G
)
grpinvinv.n  |-  N  =  ( inv g `  G )
grpinv11.g  |-  ( ph  ->  G  e.  Grp )
Assertion
Ref Expression
grpinvf1o  |-  ( ph  ->  N : B -1-1-onto-> B )

Proof of Theorem grpinvf1o
StepHypRef Expression
1 grpinv11.g . . . 4  |-  ( ph  ->  G  e.  Grp )
2 grpinvinv.b . . . . 5  |-  B  =  ( Base `  G
)
3 grpinvinv.n . . . . 5  |-  N  =  ( inv g `  G )
42, 3grpinvf 14526 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
51, 4syl 15 . . 3  |-  ( ph  ->  N : B --> B )
6 ffn 5389 . . 3  |-  ( N : B --> B  ->  N  Fn  B )
75, 6syl 15 . 2  |-  ( ph  ->  N  Fn  B )
82, 3grpinvcnv 14536 . . . . 5  |-  ( G  e.  Grp  ->  `' N  =  N )
91, 8syl 15 . . . 4  |-  ( ph  ->  `' N  =  N
)
109fneq1d 5335 . . 3  |-  ( ph  ->  ( `' N  Fn  B 
<->  N  Fn  B ) )
117, 10mpbird 223 . 2  |-  ( ph  ->  `' N  Fn  B
)
12 dff1o4 5480 . 2  |-  ( N : B -1-1-onto-> B  <->  ( N  Fn  B  /\  `' N  Fn  B ) )
137, 11, 12sylanbrc 645 1  |-  ( ph  ->  N : B -1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   `'ccnv 4688    Fn wfn 5250   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255   Basecbs 13148   Grpcgrp 14362   inv gcminusg 14363
This theorem is referenced by:  invoppggim  14833  gsumsub  15219  dprdfsub  15256  psrnegcl  16141  psrlinv  16142  lflnegl  29266
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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-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-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490
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