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Theorem subgid 19091
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 4000 . 2 (𝐺 ∈ Grp → 𝐵𝐵)
3 issubg.b . . . 4 𝐵 = (Base‘𝐺)
43ressid 17228 . . 3 (𝐺 ∈ Grp → (𝐺s 𝐵) = 𝐺)
54, 1eqeltrd 2825 . 2 (𝐺 ∈ Grp → (𝐺s 𝐵) ∈ Grp)
63issubg 19089 . 2 (𝐵 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝐵𝐵 ∧ (𝐺s 𝐵) ∈ Grp))
71, 2, 5, 6syl3anbrc 1340 1 (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wss 3944  cfv 6549  (class class class)co 7419  Basecbs 17183  s cress 17212  Grpcgrp 18898  SubGrpcsubg 19083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-ress 17213  df-subg 19086
This theorem is referenced by:  trivsubgsnd  19117  nsgid  19133  gaid2  19266  pgpfac1  20049  pgpfac  20053  ablfaclem2  20055  ablfac  20057  qusxpid  33174  qusrn  33221
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