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Theorem grpoidcl 21795
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 21794 . 2  |-  ( G  e.  GrpOp  ->  U  =  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  x ) )
41grpoideu 21787 . . 3  |-  ( G  e.  GrpOp  ->  E! u  e.  X  A. x  e.  X  ( u G x )  =  x )
5 riotacl 6556 . . 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 16 . 2  |-  ( G  e.  GrpOp  ->  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  x )  e.  X
)
73, 6eqeltrd 2509 1  |-  ( G  e.  GrpOp  ->  U  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2697   E!wreu 2699   ran crn 4871   ` cfv 5446  (class class class)co 6073   iota_crio 6534   GrpOpcgr 21764  GIdcgi 21765
This theorem is referenced by:  grpoid  21801  grpoinvid  21810  grpo2grp  21812  gxcl  21843  gxid  21851  gxdi  21874  subgoid  21885  gidsn  21926  ghomid  21943  ghgrp  21946  rngo0cl  21976  rngolz  21979  rngorz  21980  vczcl  22035  nvzcl  22105  ghomgrpilem2  25087  ghomf1olem  25095  grpokerinj  26514  keridl  26596
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-riota 6541  df-grpo 21769  df-gid 21770
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