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Theorem grpoinvf 21828
Description: Mapping of the inverse function of a group. (Contributed by NM, 29-Mar-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpasscan1.1  |-  X  =  ran  G
grpasscan1.2  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
grpoinvf  |-  ( G  e.  GrpOp  ->  N : X
-1-1-onto-> X )

Proof of Theorem grpoinvf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6553 . . . 4  |-  ( iota_ y  e.  X ( y G x )  =  (GId `  G )
)  e.  _V
2 eqid 2436 . . . 4  |-  ( x  e.  X  |->  ( iota_ y  e.  X ( y G x )  =  (GId `  G )
) )  =  ( x  e.  X  |->  (
iota_ y  e.  X
( y G x )  =  (GId `  G ) ) )
31, 2fnmpti 5573 . . 3  |-  ( x  e.  X  |->  ( iota_ y  e.  X ( y G x )  =  (GId `  G )
) )  Fn  X
4 grpasscan1.1 . . . . 5  |-  X  =  ran  G
5 eqid 2436 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
6 grpasscan1.2 . . . . 5  |-  N  =  ( inv `  G
)
74, 5, 6grpoinvfval 21812 . . . 4  |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X
( y G x )  =  (GId `  G ) ) ) )
87fneq1d 5536 . . 3  |-  ( G  e.  GrpOp  ->  ( N  Fn  X  <->  ( x  e.  X  |->  ( iota_ y  e.  X ( y G x )  =  (GId
`  G ) ) )  Fn  X ) )
93, 8mpbiri 225 . 2  |-  ( G  e.  GrpOp  ->  N  Fn  X )
10 fnrnfv 5773 . . . 4  |-  ( N  Fn  X  ->  ran  N  =  { y  |  E. x  e.  X  y  =  ( N `  x ) } )
119, 10syl 16 . . 3  |-  ( G  e.  GrpOp  ->  ran  N  =  { y  |  E. x  e.  X  y  =  ( N `  x ) } )
124, 6grpoinvcl 21814 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  ( N `  y )  e.  X )
134, 6grpo2inv 21827 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  ( N `  ( N `  y ) )  =  y )
1413eqcomd 2441 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  y  =  ( N `  ( N `  y ) ) )
15 fveq2 5728 . . . . . . . . 9  |-  ( x  =  ( N `  y )  ->  ( N `  x )  =  ( N `  ( N `  y ) ) )
1615eqeq2d 2447 . . . . . . . 8  |-  ( x  =  ( N `  y )  ->  (
y  =  ( N `
 x )  <->  y  =  ( N `  ( N `
 y ) ) ) )
1716rspcev 3052 . . . . . . 7  |-  ( ( ( N `  y
)  e.  X  /\  y  =  ( N `  ( N `  y
) ) )  ->  E. x  e.  X  y  =  ( N `  x ) )
1812, 14, 17syl2anc 643 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  E. x  e.  X  y  =  ( N `  x ) )
1918ex 424 . . . . 5  |-  ( G  e.  GrpOp  ->  ( y  e.  X  ->  E. x  e.  X  y  =  ( N `  x ) ) )
20 simpr 448 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  y  =  ( N `  x ) )  ->  y  =  ( N `  x ) )
214, 6grpoinvcl 21814 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  ( N `  x )  e.  X )
2221adantr 452 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  y  =  ( N `  x ) )  ->  ( N `  x )  e.  X
)
2320, 22eqeltrd 2510 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  y  =  ( N `  x ) )  ->  y  e.  X )
2423exp31 588 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( x  e.  X  ->  ( y  =  ( N `  x )  ->  y  e.  X ) ) )
2524rexlimdv 2829 . . . . 5  |-  ( G  e.  GrpOp  ->  ( E. x  e.  X  y  =  ( N `  x )  ->  y  e.  X ) )
2619, 25impbid 184 . . . 4  |-  ( G  e.  GrpOp  ->  ( y  e.  X  <->  E. x  e.  X  y  =  ( N `  x ) ) )
2726abbi2dv 2551 . . 3  |-  ( G  e.  GrpOp  ->  X  =  { y  |  E. x  e.  X  y  =  ( N `  x ) } )
2811, 27eqtr4d 2471 . 2  |-  ( G  e.  GrpOp  ->  ran  N  =  X )
29 fveq2 5728 . . . 4  |-  ( ( N `  x )  =  ( N `  y )  ->  ( N `  ( N `  x ) )  =  ( N `  ( N `  y )
) )
304, 6grpo2inv 21827 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  ( N `  ( N `  x ) )  =  x )
3130, 13eqeqan12d 2451 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  ( G  e. 
GrpOp  /\  y  e.  X
) )  ->  (
( N `  ( N `  x )
)  =  ( N `
 ( N `  y ) )  <->  x  =  y ) )
3231anandis 804 . . . 4  |-  ( ( G  e.  GrpOp  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( ( N `  ( N `  x ) )  =  ( N `  ( N `  y )
)  <->  x  =  y
) )
3329, 32syl5ib 211 . . 3  |-  ( ( G  e.  GrpOp  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( ( N `  x )  =  ( N `  y )  ->  x  =  y ) )
3433ralrimivva 2798 . 2  |-  ( G  e.  GrpOp  ->  A. x  e.  X  A. y  e.  X  ( ( N `  x )  =  ( N `  y )  ->  x  =  y ) )
35 dff1o6 6013 . 2  |-  ( N : X -1-1-onto-> X  <->  ( N  Fn  X  /\  ran  N  =  X  /\  A. x  e.  X  A. y  e.  X  ( ( N `  x )  =  ( N `  y )  ->  x  =  y ) ) )
369, 28, 34, 35syl3anbrc 1138 1  |-  ( G  e.  GrpOp  ->  N : X
-1-1-onto-> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705   E.wrex 2706    e. cmpt 4266   ran crn 4879    Fn wfn 5449   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   iota_crio 6542   GrpOpcgr 21774  GIdcgi 21775   invcgn 21776
This theorem is referenced by:  ginvsn  21937  nvinvfval  22121
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-pr 4403  ax-un 4701
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-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-grpo 21779  df-gid 21780  df-ginv 21781
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