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Theorem issubg 18755
Description: The subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypothesis
Ref Expression
issubg.b 𝐵 = (Base‘𝐺)
Assertion
Ref Expression
issubg (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))

Proof of Theorem issubg
Dummy variables 𝑤 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-subg 18752 . . 3 SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
21mptrcl 6884 . 2 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3 simp1 1135 . 2 ((𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp) → 𝐺 ∈ Grp)
4 fveq2 6774 . . . . . . . . . 10 (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
5 issubg.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
64, 5eqtr4di 2796 . . . . . . . . 9 (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
76pweqd 4552 . . . . . . . 8 (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
8 oveq1 7282 . . . . . . . . 9 (𝑤 = 𝐺 → (𝑤s 𝑠) = (𝐺s 𝑠))
98eleq1d 2823 . . . . . . . 8 (𝑤 = 𝐺 → ((𝑤s 𝑠) ∈ Grp ↔ (𝐺s 𝑠) ∈ Grp))
107, 9rabeqbidv 3420 . . . . . . 7 (𝑤 = 𝐺 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp} = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp})
115fvexi 6788 . . . . . . . . 9 𝐵 ∈ V
1211pwex 5303 . . . . . . . 8 𝒫 𝐵 ∈ V
1312rabex 5256 . . . . . . 7 {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp} ∈ V
1410, 1, 13fvmpt 6875 . . . . . 6 (𝐺 ∈ Grp → (SubGrp‘𝐺) = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp})
1514eleq2d 2824 . . . . 5 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp}))
16 oveq2 7283 . . . . . . . 8 (𝑠 = 𝑆 → (𝐺s 𝑠) = (𝐺s 𝑆))
1716eleq1d 2823 . . . . . . 7 (𝑠 = 𝑆 → ((𝐺s 𝑠) ∈ Grp ↔ (𝐺s 𝑆) ∈ Grp))
1817elrab 3624 . . . . . 6 (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp} ↔ (𝑆 ∈ 𝒫 𝐵 ∧ (𝐺s 𝑆) ∈ Grp))
1911elpw2 5269 . . . . . . 7 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
2019anbi1i 624 . . . . . 6 ((𝑆 ∈ 𝒫 𝐵 ∧ (𝐺s 𝑆) ∈ Grp) ↔ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))
2118, 20bitri 274 . . . . 5 (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp} ↔ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))
2215, 21bitrdi 287 . . . 4 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp)))
23 ibar 529 . . . 4 (𝐺 ∈ Grp → ((𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))))
2422, 23bitrd 278 . . 3 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))))
25 3anass 1094 . . 3 ((𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp)))
2624, 25bitr4di 289 . 2 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp)))
272, 3, 26pm5.21nii 380 1 (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  {crab 3068  wss 3887  𝒫 cpw 4533  cfv 6433  (class class class)co 7275  Basecbs 16912  s cress 16941  Grpcgrp 18577  SubGrpcsubg 18749
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-subg 18752
This theorem is referenced by:  subgss  18756  subgid  18757  subggrp  18758  subgrcl  18760  issubg2  18770  resgrpisgrp  18776  subsubg  18778  pgrpsubgsymgbi  19016  opprsubg  19878  subrgsubg  20030  subdrgint  20071  cphsubrglem  24341  suborng  31514
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