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Theorem gidval 24891
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 3122 . 2  |-  ( G  e.  V  ->  G  e.  _V )
2 rneq 5226 . . . . 5  |-  ( g  =  G  ->  ran  g  =  ran  G )
3 gidval.1 . . . . 5  |-  X  =  ran  G
42, 3syl6eqr 2526 . . . 4  |-  ( g  =  G  ->  ran  g  =  X )
5 oveq 6288 . . . . . . 7  |-  ( g  =  G  ->  (
u g x )  =  ( u G x ) )
65eqeq1d 2469 . . . . . 6  |-  ( g  =  G  ->  (
( u g x )  =  x  <->  ( u G x )  =  x ) )
7 oveq 6288 . . . . . . 7  |-  ( g  =  G  ->  (
x g u )  =  ( x G u ) )
87eqeq1d 2469 . . . . . 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 3072 . . . 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 6246 . . 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 24870 . . 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 6247 . . 3  |-  ( iota_ u  e.  X  A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x ) )  e.  _V
1411, 12, 13fvmpt 5948 . 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 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   ran crn 5000   ` cfv 5586   iota_crio 6242  (class class class)co 6282  GIdcgi 24865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fv 5594  df-riota 6243  df-ov 6285  df-gid 24870
This theorem is referenced by:  grpoidval  24894  idrval  25005  exidresid  29944
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