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Theorem grpoidcl 30202
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 30201 . 2 (𝐺 ∈ GrpOp → 𝑈 = (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥))
41grpoideu 30197 . . 3 (𝐺 ∈ GrpOp → ∃!𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
5 riotacl 7386 . . 3 (∃!𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥) ∈ 𝑋)
64, 5syl 17 . 2 (𝐺 ∈ GrpOp → (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥) ∈ 𝑋)
73, 6eqeltrd 2832 1 (𝐺 ∈ GrpOp → 𝑈𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  wral 3060  ∃!wreu 3373  ran crn 5677  cfv 6543  crio 7367  (class class class)co 7412  GrpOpcgr 30177  GIdcgi 30178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-riota 7368  df-ov 7415  df-grpo 30181  df-gid 30182
This theorem is referenced by:  grpoid  30208  vczcl  30260  nvzcl  30322  ghomidOLD  37224  grpokerinj  37228  rngo0cl  37254  rngolz  37257  rngorz  37258  gidsn  37287  keridl  37367
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