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Theorem grpoinvfval 21812
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 5131 . . . . 5  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
42, 3syl5eqel 2520 . . . 4  |-  ( G  e.  GrpOp  ->  X  e.  _V )
5 mptexg 5965 . . . 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 5095 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
87, 2syl6eqr 2486 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
9 oveq 6087 . . . . . . 7  |-  ( g  =  G  ->  (
y g x )  =  ( y G x ) )
10 fveq2 5728 . . . . . . . 8  |-  ( g  =  G  ->  (GId `  g )  =  (GId
`  G ) )
11 grpinvfval.2 . . . . . . . 8  |-  U  =  (GId `  G )
1210, 11syl6eqr 2486 . . . . . . 7  |-  ( g  =  G  ->  (GId `  g )  =  U )
139, 12eqeq12d 2450 . . . . . 6  |-  ( g  =  G  ->  (
( y g x )  =  (GId `  g )  <->  ( y G x )  =  U ) )
148, 13riotaeqbidv 6552 . . . . 5  |-  ( g  =  G  ->  ( iota_ y  e.  ran  g
( y g x )  =  (GId `  g ) )  =  ( iota_ y  e.  X
( y G x )  =  U ) )
158, 14mpteq12dv 4287 . . . 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 21781 . . . 4  |-  inv  =  ( g  e.  GrpOp  |->  ( x  e.  ran  g  |->  ( iota_ y  e. 
ran  g ( y g x )  =  (GId `  g )
) ) )
1715, 16fvmptg 5804 . . 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 650 . 2  |-  ( G  e.  GrpOp  ->  ( inv `  G )  =  ( x  e.  X  |->  (
iota_ y  e.  X
( y G x )  =  U ) ) )
191, 18syl5eq 2480 1  |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X
( y G x )  =  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956    e. cmpt 4266   ran crn 4879   ` cfv 5454  (class class class)co 6081   iota_crio 6542   GrpOpcgr 21774  GIdcgi 21775   invcgn 21776
This theorem is referenced by:  grpoinvval  21813  grpoinvf  21828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-riota 6549  df-ginv 21781
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