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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
Assertion
Ref Expression
gidval GId
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem gidval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2966 . 2
2 rneq 5098 . . . . 5
3 gidval.1 . . . . 5
42, 3syl6eqr 2488 . . . 4
5 oveq 6090 . . . . . . 7
65eqeq1d 2446 . . . . . 6
7 oveq 6090 . . . . . . 7
87eqeq1d 2446 . . . . . 6
96, 8anbi12d 693 . . . . 5
104, 9raleqbidv 2918 . . . 4
114, 10riotaeqbidv 6555 . . 3
12 df-gid 21785 . . 3 GId
13 riotaex 6556 . . 3
1411, 12, 13fvmpt 5809 . 2 GId
151, 14syl 16 1 GId
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  wral 2707  cvv 2958   crn 4882  cfv 5457  (class class class)co 6084  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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