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Theorem grpinvf1o 14554
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 14542 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
51, 4syl 15 . . 3  |-  ( ph  ->  N : B --> B )
6 ffn 5405 . . 3  |-  ( N : B --> B  ->  N  Fn  B )
75, 6syl 15 . 2  |-  ( ph  ->  N  Fn  B )
82, 3grpinvcnv 14552 . . . . 5  |-  ( G  e.  Grp  ->  `' N  =  N )
91, 8syl 15 . . . 4  |-  ( ph  ->  `' N  =  N
)
109fneq1d 5351 . . 3  |-  ( ph  ->  ( `' N  Fn  B 
<->  N  Fn  B ) )
117, 10mpbird 223 . 2  |-  ( ph  ->  `' N  Fn  B
)
12 dff1o4 5496 . 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 1632    e. wcel 1696   `'ccnv 4704    Fn wfn 5266   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271   Basecbs 13164   Grpcgrp 14378   inv gcminusg 14379
This theorem is referenced by:  invoppggim  14849  gsumsub  15235  dprdfsub  15272  psrnegcl  16157  psrlinv  16158  lflnegl  29888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506
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