Theorem grpoidcl 25091

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.

Step	Hyp	Ref	Expression
1		grpoidval.1	. . 3 |- X = ran G
2		grpoidval.2	. . 3 |- U = (GId ` G )
3	1, 2	grpoidval 25090	. 2 |- ( G e. GrpOp -> U = ( iota_ u e. X A. x e. X ( u G x ) = x ) )
4	1	grpoideu 25083	. . 3 |- ( G e. GrpOp -> E! u e. X A. x e. X ( u G x ) = x )
5		riotacl 6257	. . 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 )
6	4, 5	syl 16	. 2 |- ( G e. GrpOp -> ( iota_ u e. X A. x e. X ( u G x ) = x ) e. X )
7	3, 6	eqeltrd 2531	1 |- ( G e. GrpOp -> U e. X )

Syntax hints:   -> wi 4   = wceq 1383   e. wcel 1804   A.wral 2793   E!wreu 2795   ran crn 4990   ` cfv 5578   iota_crio 6241  (class class class)co 6281   GrpOpcgr 25060  GIdcgi 25061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586  df-riota 6242  df-ov 6284  df-grpo 25065  df-gid 25066

This theorem is referenced by:  grpoid  25097  grpoinvid  25106  grpo2grp  25108  gxcl  25139  gxid  25147  gxdi  25170  subgoid  25181  gidsn  25222  ghomidOLD  25239  ghgrpOLD  25242  rngo0cl  25272  rngolz  25275  rngorz  25276  vczcl  25331  nvzcl  25401  ghomgrpilem2  28899  ghomf1olem  28907  grpokerinj  30322  keridl  30404