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Theorem grpoinvf 25359
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 6162 . . . 4  |-  ( iota_ y  e.  X  ( y G x )  =  (GId `  G )
)  e.  _V
2 eqid 2382 . . . 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 5617 . . 3  |-  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  (GId `  G )
) )  Fn  X
4 grpasscan1.1 . . . . 5  |-  X  =  ran  G
5 eqid 2382 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
6 grpasscan1.2 . . . . 5  |-  N  =  ( inv `  G
)
74, 5, 6grpoinvfval 25343 . . . 4  |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  (GId `  G ) ) ) )
87fneq1d 5579 . . 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 5820 . . . 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 25345 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  ( N `  y )  e.  X )
134, 6grpo2inv 25358 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  ( N `  ( N `  y ) )  =  y )
1413eqcomd 2390 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  y  =  ( N `  ( N `  y ) ) )
15 fveq2 5774 . . . . . . . . 9  |-  ( x  =  ( N `  y )  ->  ( N `  x )  =  ( N `  ( N `  y ) ) )
1615eqeq2d 2396 . . . . . . . 8  |-  ( x  =  ( N `  y )  ->  (
y  =  ( N `
 x )  <->  y  =  ( N `  ( N `
 y ) ) ) )
1716rspcev 3135 . . . . . . 7  |-  ( ( ( N `  y
)  e.  X  /\  y  =  ( N `  ( N `  y
) ) )  ->  E. x  e.  X  y  =  ( N `  x ) )
1812, 14, 17syl2anc 659 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  E. x  e.  X  y  =  ( N `  x ) )
1918ex 432 . . . . 5  |-  ( G  e.  GrpOp  ->  ( y  e.  X  ->  E. x  e.  X  y  =  ( N `  x ) ) )
20 simpr 459 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  y  =  ( N `  x ) )  ->  y  =  ( N `  x ) )
214, 6grpoinvcl 25345 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  ( N `  x )  e.  X )
2221adantr 463 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  y  =  ( N `  x ) )  ->  ( N `  x )  e.  X
)
2320, 22eqeltrd 2470 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  y  =  ( N `  x ) )  ->  y  e.  X )
2423exp31 602 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( x  e.  X  ->  ( y  =  ( N `  x )  ->  y  e.  X ) ) )
2524rexlimdv 2872 . . . . 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 2519 . . 3  |-  ( G  e.  GrpOp  ->  X  =  { y  |  E. x  e.  X  y  =  ( N `  x ) } )
2811, 27eqtr4d 2426 . 2  |-  ( G  e.  GrpOp  ->  ran  N  =  X )
29 fveq2 5774 . . . 4  |-  ( ( N `  x )  =  ( N `  y )  ->  ( N `  ( N `  x ) )  =  ( N `  ( N `  y )
) )
304, 6grpo2inv 25358 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  ( N `  ( N `  x ) )  =  x )
3130, 13eqeqan12d 2405 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  ( G  e. 
GrpOp  /\  y  e.  X
) )  ->  (
( N `  ( N `  x )
)  =  ( N `
 ( N `  y ) )  <->  x  =  y ) )
3231anandis 828 . . . 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 2803 . 2  |-  ( G  e.  GrpOp  ->  A. x  e.  X  A. y  e.  X  ( ( N `  x )  =  ( N `  y )  ->  x  =  y ) )
35 dff1o6 6082 . 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 1178 1  |-  ( G  e.  GrpOp  ->  N : X
-1-1-onto-> X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   {cab 2367   A.wral 2732   E.wrex 2733    |-> cmpt 4425   ran crn 4914    Fn wfn 5491   -1-1-onto->wf1o 5495   ` cfv 5496   iota_crio 6157  (class class class)co 6196   GrpOpcgr 25305  GIdcgi 25306   invcgn 25307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-grpo 25310  df-gid 25311  df-ginv 25312
This theorem is referenced by:  ginvsn  25468  nvinvfval  25652
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