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Theorem grpinvf1o 16307
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  =  ( invg `  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  =  ( invg `  G )
42, 3grpinvf 16293 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
51, 4syl 16 . . 3  |-  ( ph  ->  N : B --> B )
6 ffn 5713 . . 3  |-  ( N : B --> B  ->  N  Fn  B )
75, 6syl 16 . 2  |-  ( ph  ->  N  Fn  B )
82, 3grpinvcnv 16305 . . . . 5  |-  ( G  e.  Grp  ->  `' N  =  N )
91, 8syl 16 . . . 4  |-  ( ph  ->  `' N  =  N
)
109fneq1d 5653 . . 3  |-  ( ph  ->  ( `' N  Fn  B 
<->  N  Fn  B ) )
117, 10mpbird 232 . 2  |-  ( ph  ->  `' N  Fn  B
)
12 dff1o4 5806 . 2  |-  ( N : B -1-1-onto-> B  <->  ( N  Fn  B  /\  `' N  Fn  B ) )
137, 11, 12sylanbrc 662 1  |-  ( ph  ->  N : B -1-1-onto-> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   `'ccnv 4987    Fn wfn 5565   -->wf 5566   -1-1-onto->wf1o 5569   ` cfv 5570   Basecbs 14716   Grpcgrp 16252   invgcminusg 16253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257
This theorem is referenced by:  invoppggim  16594  gsumsub  17171  dprdfsub  17256  dprdfsubOLD  17263  psrnegcl  18244  psrlinv  18245  mdetleib2  19257  lflnegl  35198
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