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Theorem idrval 23946
Description: The value of the identity element. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
idrval.1  |-  X  =  ran  G
idrval.2  |-  U  =  (GId `  G )
Assertion
Ref Expression
idrval  |-  ( G  e.  A  ->  U  =  ( iota_ u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
Distinct variable groups:    u, G, x    u, X, x
Allowed substitution hints:    A( x, u)    U( x, u)

Proof of Theorem idrval
StepHypRef Expression
1 idrval.2 . 2  |-  U  =  (GId `  G )
2 idrval.1 . . 3  |-  X  =  ran  G
32gidval 23832 . 2  |-  ( G  e.  A  ->  (GId `  G )  =  (
iota_ u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
41, 3syl5eq 2503 1  |-  ( G  e.  A  ->  U  =  ( 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 1370    e. wcel 1758   A.wral 2793   ran crn 4936   ` cfv 5513   iota_crio 6147  (class class class)co 6187  GIdcgi 23806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-iota 5476  df-fun 5515  df-fv 5521  df-riota 6148  df-ov 6190  df-gid 23811
This theorem is referenced by:  iorlid  23947  cmpidelt  23948
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