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Theorem grpinvf1o 14861
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 14849 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
51, 4syl 16 . . 3  |-  ( ph  ->  N : B --> B )
6 ffn 5591 . . 3  |-  ( N : B --> B  ->  N  Fn  B )
75, 6syl 16 . 2  |-  ( ph  ->  N  Fn  B )
82, 3grpinvcnv 14859 . . . . 5  |-  ( G  e.  Grp  ->  `' N  =  N )
91, 8syl 16 . . . 4  |-  ( ph  ->  `' N  =  N
)
109fneq1d 5536 . . 3  |-  ( ph  ->  ( `' N  Fn  B 
<->  N  Fn  B ) )
117, 10mpbird 224 . 2  |-  ( ph  ->  `' N  Fn  B
)
12 dff1o4 5682 . 2  |-  ( N : B -1-1-onto-> B  <->  ( N  Fn  B  /\  `' N  Fn  B ) )
137, 11, 12sylanbrc 646 1  |-  ( ph  ->  N : B -1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   `'ccnv 4877    Fn wfn 5449   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454   Basecbs 13469   Grpcgrp 14685   inv gcminusg 14686
This theorem is referenced by:  invoppggim  15156  gsumsub  15542  dprdfsub  15579  psrnegcl  16460  psrlinv  16461  lflnegl  29874
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-riota 6549  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813
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