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Theorem grpoinvf 23726
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 6055 . . . 4  |-  ( iota_ y  e.  X  ( y G x )  =  (GId `  G )
)  e.  _V
2 eqid 2442 . . . 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 5538 . . 3  |-  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  (GId `  G )
) )  Fn  X
4 grpasscan1.1 . . . . 5  |-  X  =  ran  G
5 eqid 2442 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
6 grpasscan1.2 . . . . 5  |-  N  =  ( inv `  G
)
74, 5, 6grpoinvfval 23710 . . . 4  |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  (GId `  G ) ) ) )
87fneq1d 5500 . . 3  |-  ( G  e.  GrpOp  ->  ( N  Fn  X  <->  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  (GId
`  G ) ) )  Fn  X ) )
93, 8mpbiri 233 . 2  |-  ( G  e.  GrpOp  ->  N  Fn  X )
10 fnrnfv 5737 . . . 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 23712 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  ( N `  y )  e.  X )
134, 6grpo2inv 23725 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  ( N `  ( N `  y ) )  =  y )
1413eqcomd 2447 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  y  =  ( N `  ( N `  y ) ) )
15 fveq2 5690 . . . . . . . . 9  |-  ( x  =  ( N `  y )  ->  ( N `  x )  =  ( N `  ( N `  y ) ) )
1615eqeq2d 2453 . . . . . . . 8  |-  ( x  =  ( N `  y )  ->  (
y  =  ( N `
 x )  <->  y  =  ( N `  ( N `
 y ) ) ) )
1716rspcev 3072 . . . . . . 7  |-  ( ( ( N `  y
)  e.  X  /\  y  =  ( N `  ( N `  y
) ) )  ->  E. x  e.  X  y  =  ( N `  x ) )
1812, 14, 17syl2anc 661 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  E. x  e.  X  y  =  ( N `  x ) )
1918ex 434 . . . . 5  |-  ( G  e.  GrpOp  ->  ( y  e.  X  ->  E. x  e.  X  y  =  ( N `  x ) ) )
20 simpr 461 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  y  =  ( N `  x ) )  ->  y  =  ( N `  x ) )
214, 6grpoinvcl 23712 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  ( N `  x )  e.  X )
2221adantr 465 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  y  =  ( N `  x ) )  ->  ( N `  x )  e.  X
)
2320, 22eqeltrd 2516 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  y  =  ( N `  x ) )  ->  y  e.  X )
2423exp31 604 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( x  e.  X  ->  ( y  =  ( N `  x )  ->  y  e.  X ) ) )
2524rexlimdv 2839 . . . . 5  |-  ( G  e.  GrpOp  ->  ( E. x  e.  X  y  =  ( N `  x )  ->  y  e.  X ) )
2619, 25impbid 191 . . . 4  |-  ( G  e.  GrpOp  ->  ( y  e.  X  <->  E. x  e.  X  y  =  ( N `  x ) ) )
2726abbi2dv 2557 . . 3  |-  ( G  e.  GrpOp  ->  X  =  { y  |  E. x  e.  X  y  =  ( N `  x ) } )
2811, 27eqtr4d 2477 . 2  |-  ( G  e.  GrpOp  ->  ran  N  =  X )
29 fveq2 5690 . . . 4  |-  ( ( N `  x )  =  ( N `  y )  ->  ( N `  ( N `  x ) )  =  ( N `  ( N `  y )
) )
304, 6grpo2inv 23725 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  ( N `  ( N `  x ) )  =  x )
3130, 13eqeqan12d 2457 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  ( G  e. 
GrpOp  /\  y  e.  X
) )  ->  (
( N `  ( N `  x )
)  =  ( N `
 ( N `  y ) )  <->  x  =  y ) )
3231anandis 826 . . . 4  |-  ( ( G  e.  GrpOp  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( ( N `  ( N `  x ) )  =  ( N `  ( N `  y )
)  <->  x  =  y
) )
3329, 32syl5ib 219 . . 3  |-  ( ( G  e.  GrpOp  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( ( N `  x )  =  ( N `  y )  ->  x  =  y ) )
3433ralrimivva 2807 . 2  |-  ( G  e.  GrpOp  ->  A. x  e.  X  A. y  e.  X  ( ( N `  x )  =  ( N `  y )  ->  x  =  y ) )
35 dff1o6 5981 . 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 1172 1  |-  ( G  e.  GrpOp  ->  N : X
-1-1-onto-> X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2428   A.wral 2714   E.wrex 2715    e. cmpt 4349   ran crn 4840    Fn wfn 5412   -1-1-onto->wf1o 5416   ` cfv 5417   iota_crio 6050  (class class class)co 6090   GrpOpcgr 23672  GIdcgi 23673   invcgn 23674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-grpo 23677  df-gid 23678  df-ginv 23679
This theorem is referenced by:  ginvsn  23835  nvinvfval  24019
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