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Theorem idrval 21915
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 21801 . 2  |-  ( G  e.  A  ->  (GId `  G )  =  (
iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
41, 3syl5eq 2480 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   ran crn 4879   ` cfv 5454  (class class class)co 6081   iota_crio 6542  GIdcgi 21775
This theorem is referenced by:  iorlid  21916  cmpidelt  21917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-riota 6549  df-gid 21780
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