Theorem grpoidcl 25417

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 25416	. 2 |- ( G e. GrpOp -> U = ( iota_ u e. X A. x e. X ( u G x ) = x ) )
4	1	grpoideu 25409	. . 3 |- ( G e. GrpOp -> E! u e. X A. x e. X ( u G x ) = x )
5		riotacl 6246	. . 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 2542	1 |- ( G e. GrpOp -> U e. X )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 1823   A.wral 2804   E!wreu 2806   ran crn 4989   ` cfv 5570   iota_crio 6231  (class class class)co 6270   GrpOpcgr 25386  GIdcgi 25387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-riota 6232  df-ov 6273  df-grpo 25391  df-gid 25392

This theorem is referenced by:  grpoid  25423  grpoinvid  25432  grpo2grp  25434  gxcl  25465  gxid  25473  gxdi  25496  subgoid  25507  gidsn  25548  ghomidOLD  25565  ghgrpOLD  25568  rngo0cl  25598  rngolz  25601  rngorz  25602  vczcl  25657  nvzcl  25727  ghomgrpilem2  29290  ghomf1olem  29298  grpokerinj  30587  keridl  30669