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Theorem grpinvf1o 15698
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 15684 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
51, 4syl 16 . . 3  |-  ( ph  ->  N : B --> B )
6 ffn 5657 . . 3  |-  ( N : B --> B  ->  N  Fn  B )
75, 6syl 16 . 2  |-  ( ph  ->  N  Fn  B )
82, 3grpinvcnv 15696 . . . . 5  |-  ( G  e.  Grp  ->  `' N  =  N )
91, 8syl 16 . . . 4  |-  ( ph  ->  `' N  =  N
)
109fneq1d 5599 . . 3  |-  ( ph  ->  ( `' N  Fn  B 
<->  N  Fn  B ) )
117, 10mpbird 232 . 2  |-  ( ph  ->  `' N  Fn  B
)
12 dff1o4 5747 . 2  |-  ( N : B -1-1-onto-> B  <->  ( N  Fn  B  /\  `' N  Fn  B ) )
137, 11, 12sylanbrc 664 1  |-  ( ph  ->  N : B -1-1-onto-> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   `'ccnv 4937    Fn wfn 5511   -->wf 5512   -1-1-onto->wf1o 5515   ` cfv 5516   Basecbs 14276   Grpcgrp 15512   invgcminusg 15513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-0g 14482  df-mnd 15517  df-grp 15647  df-minusg 15648
This theorem is referenced by:  invoppggim  15977  gsumsub  16552  gsumsubOLD  16553  dprdfsub  16616  dprdfsubOLD  16623  psrnegcl  17573  psrlinv  17574  mdetleib2  18510  lflnegl  33027
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