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Theorem issubg 18258
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 18255 . . 3 SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
21mptrcl 6753 . 2 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3 simp1 1132 . 2 ((𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp) → 𝐺 ∈ Grp)
4 fveq2 6646 . . . . . . . . . 10 (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
5 issubg.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
64, 5syl6eqr 2873 . . . . . . . . 9 (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
76pweqd 4534 . . . . . . . 8 (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
8 oveq1 7140 . . . . . . . . 9 (𝑤 = 𝐺 → (𝑤s 𝑠) = (𝐺s 𝑠))
98eleq1d 2895 . . . . . . . 8 (𝑤 = 𝐺 → ((𝑤s 𝑠) ∈ Grp ↔ (𝐺s 𝑠) ∈ Grp))
107, 9rabeqbidv 3464 . . . . . . 7 (𝑤 = 𝐺 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp} = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp})
115fvexi 6660 . . . . . . . . 9 𝐵 ∈ V
1211pwex 5257 . . . . . . . 8 𝒫 𝐵 ∈ V
1312rabex 5211 . . . . . . 7 {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp} ∈ V
1410, 1, 13fvmpt 6744 . . . . . 6 (𝐺 ∈ Grp → (SubGrp‘𝐺) = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp})
1514eleq2d 2896 . . . . 5 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp}))
16 oveq2 7141 . . . . . . . 8 (𝑠 = 𝑆 → (𝐺s 𝑠) = (𝐺s 𝑆))
1716eleq1d 2895 . . . . . . 7 (𝑠 = 𝑆 → ((𝐺s 𝑠) ∈ Grp ↔ (𝐺s 𝑆) ∈ Grp))
1817elrab 3660 . . . . . 6 (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp} ↔ (𝑆 ∈ 𝒫 𝐵 ∧ (𝐺s 𝑆) ∈ Grp))
1911elpw2 5224 . . . . . . 7 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
2019anbi1i 625 . . . . . 6 ((𝑆 ∈ 𝒫 𝐵 ∧ (𝐺s 𝑆) ∈ Grp) ↔ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))
2118, 20bitri 277 . . . . 5 (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp} ↔ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))
2215, 21syl6bb 289 . . . 4 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp)))
23 ibar 531 . . . 4 (𝐺 ∈ Grp → ((𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))))
2422, 23bitrd 281 . . 3 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))))
25 3anass 1091 . . 3 ((𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp)))
2624, 25syl6bbr 291 . 2 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp)))
272, 3, 26pm5.21nii 382 1 (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  {crab 3129  wss 3913  𝒫 cpw 4515  cfv 6331  (class class class)co 7133  Basecbs 16462  s cress 16463  Grpcgrp 18082  SubGrpcsubg 18252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fv 6339  df-ov 7136  df-subg 18255
This theorem is referenced by:  subgss  18259  subgid  18260  subggrp  18261  subgrcl  18263  issubg2  18273  resgrpisgrp  18279  subsubg  18281  pgrpsubgsymgbi  18515  opprsubg  19365  subrgsubg  19517  subdrgint  19558  cphsubrglem  23761  suborng  30896
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