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Theorem gidval 20880
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 2796 . 2  |-  ( G  e.  V  ->  G  e.  _V )
2 rneq 4904 . . . . 5  |-  ( g  =  G  ->  ran  g  =  ran  G )
3 gidval.1 . . . . 5  |-  X  =  ran  G
42, 3syl6eqr 2333 . . . 4  |-  ( g  =  G  ->  ran  g  =  X )
5 oveq 5864 . . . . . . 7  |-  ( g  =  G  ->  (
u g x )  =  ( u G x ) )
65eqeq1d 2291 . . . . . 6  |-  ( g  =  G  ->  (
( u g x )  =  x  <->  ( u G x )  =  x ) )
7 oveq 5864 . . . . . . 7  |-  ( g  =  G  ->  (
x g u )  =  ( x G u ) )
87eqeq1d 2291 . . . . . 6  |-  ( g  =  G  ->  (
( x g u )  =  x  <->  ( x G u )  =  x ) )
96, 8anbi12d 691 . . . . 5  |-  ( g  =  G  ->  (
( ( u g x )  =  x  /\  ( x g u )  =  x )  <->  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
104, 9raleqbidv 2748 . . . 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 6307 . . 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 20859 . . 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 6308 . . 3  |-  ( iota_ u  e.  X A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x ) )  e.  _V
1411, 12, 13fvmpt 5602 . 2  |-  ( G  e.  _V  ->  (GId `  G )  =  (
iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
151, 14syl 15 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   ran crn 4690   ` cfv 5255  (class class class)co 5858   iota_crio 6297  GIdcgi 20854
This theorem is referenced by:  grpoidval  20883  idrval  20994  rngounval2  25425  exidresid  26569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-riota 6304  df-gid 20859
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