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Theorem grpoinvf 24906
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 6242 . . . 4  |-  ( iota_ y  e.  X  ( y G x )  =  (GId `  G )
)  e.  _V
2 eqid 2462 . . . 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 5702 . . 3  |-  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  (GId `  G )
) )  Fn  X
4 grpasscan1.1 . . . . 5  |-  X  =  ran  G
5 eqid 2462 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
6 grpasscan1.2 . . . . 5  |-  N  =  ( inv `  G
)
74, 5, 6grpoinvfval 24890 . . . 4  |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  (GId `  G ) ) ) )
87fneq1d 5664 . . 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 5907 . . . 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 24892 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  ( N `  y )  e.  X )
134, 6grpo2inv 24905 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  ( N `  ( N `  y ) )  =  y )
1413eqcomd 2470 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  y  =  ( N `  ( N `  y ) ) )
15 fveq2 5859 . . . . . . . . 9  |-  ( x  =  ( N `  y )  ->  ( N `  x )  =  ( N `  ( N `  y ) ) )
1615eqeq2d 2476 . . . . . . . 8  |-  ( x  =  ( N `  y )  ->  (
y  =  ( N `
 x )  <->  y  =  ( N `  ( N `
 y ) ) ) )
1716rspcev 3209 . . . . . . 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 24892 . . . . . . . . 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 2550 . . . . . . 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 2948 . . . . 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 2599 . . 3  |-  ( G  e.  GrpOp  ->  X  =  { y  |  E. x  e.  X  y  =  ( N `  x ) } )
2811, 27eqtr4d 2506 . 2  |-  ( G  e.  GrpOp  ->  ran  N  =  X )
29 fveq2 5859 . . . 4  |-  ( ( N `  x )  =  ( N `  y )  ->  ( N `  ( N `  x ) )  =  ( N `  ( N `  y )
) )
304, 6grpo2inv 24905 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  ( N `  ( N `  x ) )  =  x )
3130, 13eqeqan12d 2485 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  ( G  e. 
GrpOp  /\  y  e.  X
) )  ->  (
( N `  ( N `  x )
)  =  ( N `
 ( N `  y ) )  <->  x  =  y ) )
3231anandis 827 . . . 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 2880 . 2  |-  ( G  e.  GrpOp  ->  A. x  e.  X  A. y  e.  X  ( ( N `  x )  =  ( N `  y )  ->  x  =  y ) )
35 dff1o6 6162 . 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 1175 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 1374    e. wcel 1762   {cab 2447   A.wral 2809   E.wrex 2810    |-> cmpt 4500   ran crn 4995    Fn wfn 5576   -1-1-onto->wf1o 5580   ` cfv 5581   iota_crio 6237  (class class class)co 6277   GrpOpcgr 24852  GIdcgi 24853   invcgn 24854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-grpo 24857  df-gid 24858  df-ginv 24859
This theorem is referenced by:  ginvsn  25015  nvinvfval  25199
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