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Theorem grpoinvfval 23709
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 6508 . . . . 5  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
42, 3syl5eqel 2525 . . . 4  |-  ( G  e.  GrpOp  ->  X  e.  _V )
5 mptexg 5945 . . . 4  |-  ( X  e.  _V  ->  (
x  e.  X  |->  (
iota_ y  e.  X  ( y G x )  =  U ) )  e.  _V )
64, 5syl 16 . . 3  |-  ( G  e.  GrpOp  ->  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  U ) )  e. 
_V )
7 rneq 5063 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
87, 2syl6eqr 2491 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
9 oveq 6095 . . . . . . 7  |-  ( g  =  G  ->  (
y g x )  =  ( y G x ) )
10 fveq2 5689 . . . . . . . 8  |-  ( g  =  G  ->  (GId `  g )  =  (GId
`  G ) )
11 grpinvfval.2 . . . . . . . 8  |-  U  =  (GId `  G )
1210, 11syl6eqr 2491 . . . . . . 7  |-  ( g  =  G  ->  (GId `  g )  =  U )
139, 12eqeq12d 2455 . . . . . 6  |-  ( g  =  G  ->  (
( y g x )  =  (GId `  g )  <->  ( y G x )  =  U ) )
148, 13riotaeqbidv 6053 . . . . 5  |-  ( g  =  G  ->  ( iota_ y  e.  ran  g
( y g x )  =  (GId `  g ) )  =  ( iota_ y  e.  X  ( y G x )  =  U ) )
158, 14mpteq12dv 4368 . . . 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 23678 . . . 4  |-  inv  =  ( g  e.  GrpOp  |->  ( x  e.  ran  g  |->  ( iota_ y  e. 
ran  g ( y g x )  =  (GId `  g )
) ) )
1715, 16fvmptg 5770 . . 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 668 . 2  |-  ( G  e.  GrpOp  ->  ( inv `  G )  =  ( x  e.  X  |->  (
iota_ y  e.  X  ( y G x )  =  U ) ) )
191, 18syl5eq 2485 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 1369    e. wcel 1756   _Vcvv 2970    e. cmpt 4348   ran crn 4839   ` cfv 5416   iota_crio 6049  (class class class)co 6089   GrpOpcgr 23671  GIdcgi 23672   invcgn 23673
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pr 4529  ax-un 6370
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-ginv 23678
This theorem is referenced by:  grpoinvval  23710  grpoinvf  23725
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