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Theorem gidval 21806
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 2966 . 2  |-  ( G  e.  V  ->  G  e.  _V )
2 rneq 5098 . . . . 5  |-  ( g  =  G  ->  ran  g  =  ran  G )
3 gidval.1 . . . . 5  |-  X  =  ran  G
42, 3syl6eqr 2488 . . . 4  |-  ( g  =  G  ->  ran  g  =  X )
5 oveq 6090 . . . . . . 7  |-  ( g  =  G  ->  (
u g x )  =  ( u G x ) )
65eqeq1d 2446 . . . . . 6  |-  ( g  =  G  ->  (
( u g x )  =  x  <->  ( u G x )  =  x ) )
7 oveq 6090 . . . . . . 7  |-  ( g  =  G  ->  (
x g u )  =  ( x G u ) )
87eqeq1d 2446 . . . . . 6  |-  ( g  =  G  ->  (
( x g u )  =  x  <->  ( x G u )  =  x ) )
96, 8anbi12d 693 . . . . 5  |-  ( g  =  G  ->  (
( ( u g x )  =  x  /\  ( x g u )  =  x )  <->  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
104, 9raleqbidv 2918 . . . 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 6555 . . 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 21785 . . 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 6556 . . 3  |-  ( iota_ u  e.  X A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x ) )  e.  _V
1411, 12, 13fvmpt 5809 . 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958   ran crn 4882   ` cfv 5457  (class class class)co 6084   iota_crio 6545  GIdcgi 21780
This theorem is referenced by:  grpoidval  21809  idrval  21920  exidresid  26568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-riota 6552  df-gid 21785
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