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Theorem gidval 23700
Description: The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
gidval.1  |-  X  =  ran  G
Assertion
Ref Expression
gidval  |-  ( G  e.  V  ->  (GId `  G )  =  (
iota_ u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
Distinct variable groups:    x, u, G    u, X, x
Allowed substitution hints:    V( x, u)

Proof of Theorem gidval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 elex 2981 . 2  |-  ( G  e.  V  ->  G  e.  _V )
2 rneq 5065 . . . . 5  |-  ( g  =  G  ->  ran  g  =  ran  G )
3 gidval.1 . . . . 5  |-  X  =  ran  G
42, 3syl6eqr 2493 . . . 4  |-  ( g  =  G  ->  ran  g  =  X )
5 oveq 6097 . . . . . . 7  |-  ( g  =  G  ->  (
u g x )  =  ( u G x ) )
65eqeq1d 2451 . . . . . 6  |-  ( g  =  G  ->  (
( u g x )  =  x  <->  ( u G x )  =  x ) )
7 oveq 6097 . . . . . . 7  |-  ( g  =  G  ->  (
x g u )  =  ( x G u ) )
87eqeq1d 2451 . . . . . 6  |-  ( g  =  G  ->  (
( x g u )  =  x  <->  ( x G u )  =  x ) )
96, 8anbi12d 710 . . . . 5  |-  ( g  =  G  ->  (
( ( u g x )  =  x  /\  ( x g u )  =  x )  <->  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
104, 9raleqbidv 2931 . . . 4  |-  ( g  =  G  ->  ( A. x  e.  ran  g ( ( u g x )  =  x  /\  ( x g u )  =  x )  <->  A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x ) ) )
114, 10riotaeqbidv 6055 . . 3  |-  ( g  =  G  ->  ( iota_ u  e.  ran  g A. x  e.  ran  g ( ( u g x )  =  x  /\  ( x g u )  =  x ) )  =  ( iota_ u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
12 df-gid 23679 . . 3  |- GId  =  ( g  e.  _V  |->  (
iota_ u  e.  ran  g A. x  e.  ran  g ( ( u g x )  =  x  /\  ( x g u )  =  x ) ) )
13 riotaex 6056 . . 3  |-  ( iota_ u  e.  X  A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x ) )  e.  _V
1411, 12, 13fvmpt 5774 . 2  |-  ( G  e.  _V  ->  (GId `  G )  =  (
iota_ u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
151, 14syl 16 1  |-  ( G  e.  V  ->  (GId `  G )  =  (
iota_ u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972   ran crn 4841   ` cfv 5418   iota_crio 6051  (class class class)co 6091  GIdcgi 23674
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fv 5426  df-riota 6052  df-ov 6094  df-gid 23679
This theorem is referenced by:  grpoidval  23703  idrval  23814  exidresid  28744
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