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Theorem grpoinvf 20923
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 6324 . . . 4  |-  ( iota_ y  e.  X ( y G x )  =  (GId `  G )
)  e.  _V
2 eqid 2296 . . . 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 5388 . . 3  |-  ( x  e.  X  |->  ( iota_ y  e.  X ( y G x )  =  (GId `  G )
) )  Fn  X
4 grpasscan1.1 . . . . 5  |-  X  =  ran  G
5 eqid 2296 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
6 grpasscan1.2 . . . . 5  |-  N  =  ( inv `  G
)
74, 5, 6grpoinvfval 20907 . . . 4  |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X
( y G x )  =  (GId `  G ) ) ) )
87fneq1d 5351 . . 3  |-  ( G  e.  GrpOp  ->  ( N  Fn  X  <->  ( x  e.  X  |->  ( iota_ y  e.  X ( y G x )  =  (GId
`  G ) ) )  Fn  X ) )
93, 8mpbiri 224 . 2  |-  ( G  e.  GrpOp  ->  N  Fn  X )
10 fnrnfv 5585 . . . 4  |-  ( N  Fn  X  ->  ran  N  =  { y  |  E. x  e.  X  y  =  ( N `  x ) } )
119, 10syl 15 . . 3  |-  ( G  e.  GrpOp  ->  ran  N  =  { y  |  E. x  e.  X  y  =  ( N `  x ) } )
124, 6grpoinvcl 20909 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  ( N `  y )  e.  X )
134, 6grpo2inv 20922 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  ( N `  ( N `  y ) )  =  y )
1413eqcomd 2301 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  y  =  ( N `  ( N `  y ) ) )
15 fveq2 5541 . . . . . . . . 9  |-  ( x  =  ( N `  y )  ->  ( N `  x )  =  ( N `  ( N `  y ) ) )
1615eqeq2d 2307 . . . . . . . 8  |-  ( x  =  ( N `  y )  ->  (
y  =  ( N `
 x )  <->  y  =  ( N `  ( N `
 y ) ) ) )
1716rspcev 2897 . . . . . . 7  |-  ( ( ( N `  y
)  e.  X  /\  y  =  ( N `  ( N `  y
) ) )  ->  E. x  e.  X  y  =  ( N `  x ) )
1812, 14, 17syl2anc 642 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  E. x  e.  X  y  =  ( N `  x ) )
1918ex 423 . . . . 5  |-  ( G  e.  GrpOp  ->  ( y  e.  X  ->  E. x  e.  X  y  =  ( N `  x ) ) )
20 simpr 447 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  y  =  ( N `  x ) )  ->  y  =  ( N `  x ) )
214, 6grpoinvcl 20909 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  ( N `  x )  e.  X )
2221adantr 451 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  y  =  ( N `  x ) )  ->  ( N `  x )  e.  X
)
2320, 22eqeltrd 2370 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  y  =  ( N `  x ) )  ->  y  e.  X )
2423exp31 587 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( x  e.  X  ->  ( y  =  ( N `  x )  ->  y  e.  X ) ) )
2524rexlimdv 2679 . . . . 5  |-  ( G  e.  GrpOp  ->  ( E. x  e.  X  y  =  ( N `  x )  ->  y  e.  X ) )
2619, 25impbid 183 . . . 4  |-  ( G  e.  GrpOp  ->  ( y  e.  X  <->  E. x  e.  X  y  =  ( N `  x ) ) )
2726abbi2dv 2411 . . 3  |-  ( G  e.  GrpOp  ->  X  =  { y  |  E. x  e.  X  y  =  ( N `  x ) } )
2811, 27eqtr4d 2331 . 2  |-  ( G  e.  GrpOp  ->  ran  N  =  X )
29 fveq2 5541 . . . 4  |-  ( ( N `  x )  =  ( N `  y )  ->  ( N `  ( N `  x ) )  =  ( N `  ( N `  y )
) )
304, 6grpo2inv 20922 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  ( N `  ( N `  x ) )  =  x )
3130, 13eqeqan12d 2311 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  x  e.  X )  /\  ( G  e. 
GrpOp  /\  y  e.  X
) )  ->  (
( N `  ( N `  x )
)  =  ( N `
 ( N `  y ) )  <->  x  =  y ) )
3231anandis 803 . . . 4  |-  ( ( G  e.  GrpOp  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( ( N `  ( N `  x ) )  =  ( N `  ( N `  y )
)  <->  x  =  y
) )
3329, 32syl5ib 210 . . 3  |-  ( ( G  e.  GrpOp  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( ( N `  x )  =  ( N `  y )  ->  x  =  y ) )
3433ralrimivva 2648 . 2  |-  ( G  e.  GrpOp  ->  A. x  e.  X  A. y  e.  X  ( ( N `  x )  =  ( N `  y )  ->  x  =  y ) )
35 dff1o6 5807 . 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 1136 1  |-  ( G  e.  GrpOp  ->  N : X
-1-1-onto-> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    e. cmpt 4093   ran crn 4706    Fn wfn 5266   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   iota_crio 6313   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871
This theorem is referenced by:  ginvsn  21032  nvinvfval  21214  grpodlcan  25479  grpodivzer  25480
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-pr 4230  ax-un 4528
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-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-grpo 20874  df-gid 20875  df-ginv 20876
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