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Theorem issubg 18999
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 18996 . . 3 SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
21mptrcl 7002 . 2 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3 simp1 1137 . 2 ((𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp) → 𝐺 ∈ Grp)
4 fveq2 6887 . . . . . . . . . 10 (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
5 issubg.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
64, 5eqtr4di 2791 . . . . . . . . 9 (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
76pweqd 4617 . . . . . . . 8 (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
8 oveq1 7410 . . . . . . . . 9 (𝑤 = 𝐺 → (𝑤s 𝑠) = (𝐺s 𝑠))
98eleq1d 2819 . . . . . . . 8 (𝑤 = 𝐺 → ((𝑤s 𝑠) ∈ Grp ↔ (𝐺s 𝑠) ∈ Grp))
107, 9rabeqbidv 3450 . . . . . . 7 (𝑤 = 𝐺 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp} = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp})
115fvexi 6901 . . . . . . . . 9 𝐵 ∈ V
1211pwex 5376 . . . . . . . 8 𝒫 𝐵 ∈ V
1312rabex 5330 . . . . . . 7 {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp} ∈ V
1410, 1, 13fvmpt 6993 . . . . . 6 (𝐺 ∈ Grp → (SubGrp‘𝐺) = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp})
1514eleq2d 2820 . . . . 5 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp}))
16 oveq2 7411 . . . . . . . 8 (𝑠 = 𝑆 → (𝐺s 𝑠) = (𝐺s 𝑆))
1716eleq1d 2819 . . . . . . 7 (𝑠 = 𝑆 → ((𝐺s 𝑠) ∈ Grp ↔ (𝐺s 𝑆) ∈ Grp))
1817elrab 3681 . . . . . 6 (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺s 𝑠) ∈ Grp} ↔ (𝑆 ∈ 𝒫 𝐵 ∧ (𝐺s 𝑆) ∈ Grp))
1911elpw2 5343 . . . . . . 7 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
2019anbi1i 625 . . . . . 6 ((𝑆 ∈ 𝒫 𝐵 ∧ (𝐺s 𝑆) ∈ Grp) ↔ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))
2118, 20bitri 275 . . . . 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 530 . . . 4 (𝐺 ∈ Grp → ((𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))))
2422, 23bitrd 279 . . 3 (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ (𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))))
25 3anass 1096 . . 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 397  w3a 1088   = wceq 1542  wcel 2107  {crab 3433  wss 3946  𝒫 cpw 4600  cfv 6539  (class class class)co 7403  Basecbs 17139  s cress 17168  Grpcgrp 18814  SubGrpcsubg 18993
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fv 6547  df-ov 7406  df-subg 18996
This theorem is referenced by:  subgss  19000  subgid  19001  subggrp  19002  subgrcl  19004  issubg2  19014  resgrpisgrp  19020  subsubg  19022  pgrpsubgsymgbi  19268  opprsubg  20154  subrgsubg  20356  subdrgint  20406  cphsubrglem  24675  suborng  32401  subrngsubg  46663
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