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Theorem grpoidcl 20884
Description: The identity element of a group belongs to the group. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpoidval.1  |-  X  =  ran  G
grpoidval.2  |-  U  =  (GId `  G )
Assertion
Ref Expression
grpoidcl  |-  ( G  e.  GrpOp  ->  U  e.  X )

Proof of Theorem grpoidcl
Dummy variables  x  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpoidval.1 . . 3  |-  X  =  ran  G
2 grpoidval.2 . . 3  |-  U  =  (GId `  G )
31, 2grpoidval 20883 . 2  |-  ( G  e.  GrpOp  ->  U  =  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  x ) )
41grpoideu 20876 . . 3  |-  ( G  e.  GrpOp  ->  E! u  e.  X  A. x  e.  X  ( u G x )  =  x )
5 riotacl 6319 . . 3  |-  ( E! u  e.  X  A. x  e.  X  (
u G x )  =  x  ->  ( iota_ u  e.  X A. x  e.  X  (
u G x )  =  x )  e.  X )
64, 5syl 15 . 2  |-  ( G  e.  GrpOp  ->  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  x )  e.  X
)
73, 6eqeltrd 2357 1  |-  ( G  e.  GrpOp  ->  U  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   E!wreu 2545   ran crn 4690   ` cfv 5255  (class class class)co 5858   iota_crio 6297   GrpOpcgr 20853  GIdcgi 20854
This theorem is referenced by:  grpoid  20890  grpoinvid  20899  grpo2grp  20901  gxcl  20932  gxid  20940  gxdi  20963  subgoid  20974  gidsn  21015  ghomid  21032  ghgrp  21035  rngo0cl  21065  rngolz  21068  rngorz  21069  vczcl  21122  nvzcl  21192  ghomgrpilem2  23993  ghomf1olem  24001  grpodivone  25373  grpodivfo  25374  grpokerinj  26575  keridl  26657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-riota 6304  df-grpo 20858  df-gid 20859
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