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Theorem issubg 17860
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 17857 . . . 4 SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
21dmmptss 5817 . . 3 dom SubGrp ⊆ Grp
3 elfvdm 6407 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ dom SubGrp)
42, 3sseldi 3759 . 2 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5 simp1 1166 . 2 ((𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp) → 𝐺 ∈ Grp)
6 fveq2 6375 . . . . . . . . . 10 (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
7 issubg.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
86, 7syl6eqr 2817 . . . . . . . . 9 (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
98pweqd 4320 . . . . . . . 8 (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
10 oveq1 6849 . . . . . . . . 9 (𝑤 = 𝐺 → (𝑤s 𝑠) = (𝐺s 𝑠))
1110eleq1d 2829 . . . . . . . 8 (𝑤 = 𝐺 → ((𝑤s 𝑠) ∈ Grp ↔ (𝐺s 𝑠) ∈ Grp))
129, 11rabeqbidv 3344 . . . . . . 7 (𝑤 = 𝐺 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp} = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp})
137fvexi 6389 . . . . . . . . 9 𝐵 ∈ V
1413pwex 5016 . . . . . . . 8 𝒫 𝐵 ∈ V
1514rabex 4973 . . . . . . 7 {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp} ∈ V
1612, 1, 15fvmpt 6471 . . . . . 6 (𝐺 ∈ Grp → (SubGrp‘𝐺) = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp})
1716eleq2d 2830 . . . . 5 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp}))
18 oveq2 6850 . . . . . . . 8 (𝑠 = 𝑆 → (𝐺s 𝑠) = (𝐺s 𝑆))
1918eleq1d 2829 . . . . . . 7 (𝑠 = 𝑆 → ((𝐺s 𝑠) ∈ Grp ↔ (𝐺s 𝑆) ∈ Grp))
2019elrab 3519 . . . . . 6 (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp} ↔ (𝑆 ∈ 𝒫 𝐵 ∧ (𝐺s 𝑆) ∈ Grp))
2113elpw2 4986 . . . . . . 7 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
2221anbi1i 617 . . . . . 6 ((𝑆 ∈ 𝒫 𝐵 ∧ (𝐺s 𝑆) ∈ Grp) ↔ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))
2320, 22bitri 266 . . . . 5 (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp} ↔ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))
2417, 23syl6bb 278 . . . 4 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp)))
25 ibar 524 . . . 4 (𝐺 ∈ Grp → ((𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))))
2624, 25bitrd 270 . . 3 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))))
27 3anass 1116 . . 3 ((𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp)))
2826, 27syl6bbr 280 . 2 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp)))
294, 5, 28pm5.21nii 369 1 (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  {crab 3059  wss 3732  𝒫 cpw 4315  dom cdm 5277  cfv 6068  (class class class)co 6842  Basecbs 16132  s cress 16133  Grpcgrp 17691  SubGrpcsubg 17854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-subg 17857
This theorem is referenced by:  subgss  17861  subgid  17862  subggrp  17863  subgrcl  17865  issubg2  17875  resgrpisgrp  17881  subsubg  17883  pgrpsubgsymgbi  18092  opprsubg  18903  subrgsubg  19055  cphsubrglem  23255  suborng  30197
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