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Theorem subgid 19147
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 4006 . 2 (𝐺 ∈ Grp → 𝐵𝐵)
3 issubg.b . . . 4 𝐵 = (Base‘𝐺)
43ressid 17291 . . 3 (𝐺 ∈ Grp → (𝐺s 𝐵) = 𝐺)
54, 1eqeltrd 2840 . 2 (𝐺 ∈ Grp → (𝐺s 𝐵) ∈ Grp)
63issubg 19145 . 2 (𝐵 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝐵𝐵 ∧ (𝐺s 𝐵) ∈ Grp))
71, 2, 5, 6syl3anbrc 1343 1 (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  wss 3950  cfv 6560  (class class class)co 7432  Basecbs 17248  s cress 17275  Grpcgrp 18952  SubGrpcsubg 19139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-ress 17276  df-subg 19142
This theorem is referenced by:  trivsubgsnd  19173  nsgid  19189  gaid2  19322  pgpfac1  20101  pgpfac  20105  ablfaclem2  20107  ablfac  20109  qusxpid  33392  qusrn  33438
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