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Theorem grpoidcl 21655
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 21654 . 2  |-  ( G  e.  GrpOp  ->  U  =  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  x ) )
41grpoideu 21647 . . 3  |-  ( G  e.  GrpOp  ->  E! u  e.  X  A. x  e.  X  ( u G x )  =  x )
5 riotacl 6502 . . 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 2463 1  |-  ( G  e.  GrpOp  ->  U  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   A.wral 2651   E!wreu 2653   ran crn 4821   ` cfv 5396  (class class class)co 6022   iota_crio 6480   GrpOpcgr 21624  GIdcgi 21625
This theorem is referenced by:  grpoid  21661  grpoinvid  21670  grpo2grp  21672  gxcl  21703  gxid  21711  gxdi  21734  subgoid  21745  gidsn  21786  ghomid  21803  ghgrp  21806  rngo0cl  21836  rngolz  21839  rngorz  21840  vczcl  21895  nvzcl  21965  ghomgrpilem2  24878  ghomf1olem  24886  grpokerinj  26253  keridl  26335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fo 5402  df-fv 5404  df-ov 6025  df-riota 6487  df-grpo 21629  df-gid 21630
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