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Theorem grpoidcl 29498
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 ∈ GrpOp → 𝑈𝑋)

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 29497 . 2 (𝐺 ∈ GrpOp → 𝑈 = (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥))
41grpoideu 29493 . . 3 (𝐺 ∈ GrpOp → ∃!𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
5 riotacl 7336 . . 3 (∃!𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥) ∈ 𝑋)
64, 5syl 17 . 2 (𝐺 ∈ GrpOp → (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥) ∈ 𝑋)
73, 6eqeltrd 2838 1 (𝐺 ∈ GrpOp → 𝑈𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wral 3065  ∃!wreu 3354  ran crn 5639  cfv 6501  crio 7317  (class class class)co 7362  GrpOpcgr 29473  GIdcgi 29474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fo 6507  df-fv 6509  df-riota 7318  df-ov 7365  df-grpo 29477  df-gid 29478
This theorem is referenced by:  grpoid  29504  vczcl  29556  nvzcl  29618  ghomidOLD  36377  grpokerinj  36381  rngo0cl  36407  rngolz  36410  rngorz  36411  gidsn  36440  keridl  36520
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