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Theorem grpoinvfval 25797
Description: The inverse function of a group. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinvfval.1  |-  X  =  ran  G
grpinvfval.2  |-  U  =  (GId `  G )
grpinvfval.3  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
grpoinvfval  |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  U ) ) )
Distinct variable groups:    x, y, G    x, X, y    x, U
Allowed substitution hints:    U( y)    N( x, y)

Proof of Theorem grpoinvfval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 grpinvfval.3 . 2  |-  N  =  ( inv `  G
)
2 grpinvfval.1 . . . . 5  |-  X  =  ran  G
3 rnexg 6739 . . . . 5  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
42, 3syl5eqel 2521 . . . 4  |-  ( G  e.  GrpOp  ->  X  e.  _V )
5 mptexg 6150 . . . 4  |-  ( X  e.  _V  ->  (
x  e.  X  |->  (
iota_ y  e.  X  ( y G x )  =  U ) )  e.  _V )
64, 5syl 17 . . 3  |-  ( G  e.  GrpOp  ->  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  U ) )  e. 
_V )
7 rneq 5080 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
87, 2syl6eqr 2488 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
9 oveq 6311 . . . . . . 7  |-  ( g  =  G  ->  (
y g x )  =  ( y G x ) )
10 fveq2 5881 . . . . . . . 8  |-  ( g  =  G  ->  (GId `  g )  =  (GId
`  G ) )
11 grpinvfval.2 . . . . . . . 8  |-  U  =  (GId `  G )
1210, 11syl6eqr 2488 . . . . . . 7  |-  ( g  =  G  ->  (GId `  g )  =  U )
139, 12eqeq12d 2451 . . . . . 6  |-  ( g  =  G  ->  (
( y g x )  =  (GId `  g )  <->  ( y G x )  =  U ) )
148, 13riotaeqbidv 6270 . . . . 5  |-  ( g  =  G  ->  ( iota_ y  e.  ran  g
( y g x )  =  (GId `  g ) )  =  ( iota_ y  e.  X  ( y G x )  =  U ) )
158, 14mpteq12dv 4504 . . . 4  |-  ( g  =  G  ->  (
x  e.  ran  g  |->  ( iota_ y  e.  ran  g ( y g x )  =  (GId
`  g ) ) )  =  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  U ) ) )
16 df-ginv 25766 . . . 4  |-  inv  =  ( g  e.  GrpOp  |->  ( x  e.  ran  g  |->  ( iota_ y  e. 
ran  g ( y g x )  =  (GId `  g )
) ) )
1715, 16fvmptg 5962 . . 3  |-  ( ( G  e.  GrpOp  /\  (
x  e.  X  |->  (
iota_ y  e.  X  ( y G x )  =  U ) )  e.  _V )  ->  ( inv `  G
)  =  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  U ) ) )
186, 17mpdan 672 . 2  |-  ( G  e.  GrpOp  ->  ( inv `  G )  =  ( x  e.  X  |->  (
iota_ y  e.  X  ( y G x )  =  U ) ) )
191, 18syl5eq 2482 1  |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   _Vcvv 3087    |-> cmpt 4484   ran crn 4855   ` cfv 5601   iota_crio 6266  (class class class)co 6305   GrpOpcgr 25759  GIdcgi 25760   invcgn 25761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-ginv 25766
This theorem is referenced by:  grpoinvval  25798  grpoinvf  25813
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