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Theorem subgid 18221
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 3989 . 2 (𝐺 ∈ Grp → 𝐵𝐵)
3 issubg.b . . . 4 𝐵 = (Base‘𝐺)
43ressid 16549 . . 3 (𝐺 ∈ Grp → (𝐺s 𝐵) = 𝐺)
54, 1eqeltrd 2913 . 2 (𝐺 ∈ Grp → (𝐺s 𝐵) ∈ Grp)
63issubg 18219 . 2 (𝐵 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝐵𝐵 ∧ (𝐺s 𝐵) ∈ Grp))
71, 2, 5, 6syl3anbrc 1335 1 (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wss 3935  cfv 6349  (class class class)co 7145  Basecbs 16473  s cress 16474  Grpcgrp 18043  SubGrpcsubg 18213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-ress 16481  df-subg 18216
This theorem is referenced by:  trivsubgsnd  18246  nsgid  18262  gaid2  18373  pgpfac1  19133  pgpfac  19137  ablfaclem2  19139  ablfac  19141  qusxpid  30856
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