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Theorem subgid 18863
Description: A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypothesis
Ref Expression
issubg.b 𝐵 = (Base‘𝐺)
Assertion
Ref Expression
subgid (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))

Proof of Theorem subgid
StepHypRef Expression
1 id 22 . 2 (𝐺 ∈ Grp → 𝐺 ∈ Grp)
2 ssidd 3966 . 2 (𝐺 ∈ Grp → 𝐵𝐵)
3 issubg.b . . . 4 𝐵 = (Base‘𝐺)
43ressid 17060 . . 3 (𝐺 ∈ Grp → (𝐺s 𝐵) = 𝐺)
54, 1eqeltrd 2839 . 2 (𝐺 ∈ Grp → (𝐺s 𝐵) ∈ Grp)
63issubg 18861 . 2 (𝐵 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝐵𝐵 ∧ (𝐺s 𝐵) ∈ Grp))
71, 2, 5, 6syl3anbrc 1344 1 (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wss 3909  cfv 6492  (class class class)co 7350  Basecbs 17018  s cress 17047  Grpcgrp 18683  SubGrpcsubg 18855
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 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6444  df-fun 6494  df-fv 6500  df-ov 7353  df-oprab 7354  df-mpo 7355  df-ress 17048  df-subg 18858
This theorem is referenced by:  trivsubgsnd  18888  nsgid  18904  gaid2  19015  pgpfac1  19788  pgpfac  19792  ablfaclem2  19794  ablfac  19796  qusxpid  31932
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