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Theorem grpoidcl 20900
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 20899 . 2  |-  ( G  e.  GrpOp  ->  U  =  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  x ) )
41grpoideu 20892 . . 3  |-  ( G  e.  GrpOp  ->  E! u  e.  X  A. x  e.  X  ( u G x )  =  x )
5 riotacl 6335 . . 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 2370 1  |-  ( G  e.  GrpOp  ->  U  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   E!wreu 2558   ran crn 4706   ` cfv 5271  (class class class)co 5874   iota_crio 6313   GrpOpcgr 20869  GIdcgi 20870
This theorem is referenced by:  grpoid  20906  grpoinvid  20915  grpo2grp  20917  gxcl  20948  gxid  20956  gxdi  20979  subgoid  20990  gidsn  21031  ghomid  21048  ghgrp  21051  rngo0cl  21081  rngolz  21084  rngorz  21085  vczcl  21138  nvzcl  21208  ghomgrpilem2  24008  ghomf1olem  24016  grpodivone  25476  grpodivfo  25477  grpokerinj  26678  keridl  26760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-riota 6320  df-grpo 20874  df-gid 20875
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