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Theorem grpinvf1o 12945
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 12925 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
51, 4syl 14 . . 3  |-  ( ph  ->  N : B --> B )
65ffnd 5368 . 2  |-  ( ph  ->  N  Fn  B )
72, 3grpinvcnv 12943 . . . . 5  |-  ( G  e.  Grp  ->  `' N  =  N )
81, 7syl 14 . . . 4  |-  ( ph  ->  `' N  =  N
)
98fneq1d 5308 . . 3  |-  ( ph  ->  ( `' N  Fn  B 
<->  N  Fn  B ) )
106, 9mpbird 167 . 2  |-  ( ph  ->  `' N  Fn  B
)
11 dff1o4 5471 . 2  |-  ( N : B -1-1-onto-> B  <->  ( N  Fn  B  /\  `' N  Fn  B ) )
126, 10, 11sylanbrc 417 1  |-  ( ph  ->  N : B -1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   `'ccnv 4627    Fn wfn 5213   -->wf 5214   -1-1-onto->wf1o 5217   ` cfv 5218   Basecbs 12464   Grpcgrp 12882   invgcminusg 12883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886
This theorem is referenced by: (None)
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