Theorem issubg 19336

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 ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))

Proof of Theorem issubg

Dummy variables 𝑤 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-subg 19333	. . 3 ⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
2	1	mptrcl 7003	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3		simp1 1154	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) → 𝐺 ∈ Grp)
4		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
5		issubg.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
6	4, 5	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
7	6	pweqd 4574	. . . . . . . 8 ⊢ (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
8		oveq1 7427	. . . . . . . . 9 ⊢ (𝑤 = 𝐺 → (𝑤 ↾s 𝑠) = (𝐺 ↾s 𝑠))
9	8	eleq1d 2846	. . . . . . . 8 ⊢ (𝑤 = 𝐺 → ((𝑤 ↾s 𝑠) ∈ Grp ↔ (𝐺 ↾s 𝑠) ∈ Grp))
10	7, 9	rabeqbidv 3430	. . . . . . 7 ⊢ (𝑤 = 𝐺 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp} = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp})
11	5	fvexi 6899	. . . . . . . . 9 ⊢ 𝐵 ∈ V
12	11	pwex 5342	. . . . . . . 8 ⊢ 𝒫 𝐵 ∈ V
13	12	rabex 5300	. . . . . . 7 ⊢ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp} ∈ V
14	10, 1, 13	fvmpt 6993	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp})
15	14	eleq2d 2847	. . . . 5 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp}))
16		oveq2 7428	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝐺 ↾s 𝑠) = (𝐺 ↾s 𝑆))
17	16	eleq1d 2846	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝐺 ↾s 𝑠) ∈ Grp ↔ (𝐺 ↾s 𝑆) ∈ Grp))
18	17	elrab 3645	. . . . . 6 ⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp} ↔ (𝑆 ∈ 𝒫 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))
19	11	elpw2 5296	. . . . . . 7 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵)
20	19	anbi1i 636	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) ↔ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))
21	18, 20	bitri 278	. . . . 5 ⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp} ↔ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))
22	15, 21	bitrdi 290	. . . 4 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)))
23		ibar 538	. . . 4 ⊢ (𝐺 ∈ Grp → ((𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))))
24	22, 23	bitrd 282	. . 3 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))))
25		3anass 1111	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)))
26	24, 25	bitr4di 292	. 2 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)))
27	2, 3, 26	pm5.21nii 381	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3413   ⊆ wss 3899  𝒫 cpw 4557  ‘cfv 6538  (class class class)co 7420  Basecbs 17387   ↾_s cress 17408  Grpcgrp 19144  SubGrpcsubg 19330

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-subg 19333

This theorem is used by:  subgss  19337  subgid  19338  subggrp  19339  subgrcl  19341  issubg2  19352  resgrpisgrp  19358  subsubg  19360  pgrpsubgsymgbi  19622  opprsubg  20582  subrngsubg  20804  subrgsubg  20829  subdrgint  21060  suborng  21133  cphsubrglem  25498  algextdeglem8  34356