Theorem issubg 19096

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 19093	. . 3 ⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
2	1	mptrcl 6952	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3		simp1 1137	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) → 𝐺 ∈ Grp)
4		fveq2 6835	. . . . . . . . . 10 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
5		issubg.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
6	4, 5	eqtr4di 2790	. . . . . . . . 9 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
7	6	pweqd 4559	. . . . . . . 8 ⊢ (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
8		oveq1 7368	. . . . . . . . 9 ⊢ (𝑤 = 𝐺 → (𝑤 ↾s 𝑠) = (𝐺 ↾s 𝑠))
9	8	eleq1d 2822	. . . . . . . 8 ⊢ (𝑤 = 𝐺 → ((𝑤 ↾s 𝑠) ∈ Grp ↔ (𝐺 ↾s 𝑠) ∈ Grp))
10	7, 9	rabeqbidv 3408	. . . . . . 7 ⊢ (𝑤 = 𝐺 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp} = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp})
11	5	fvexi 6849	. . . . . . . . 9 ⊢ 𝐵 ∈ V
12	11	pwex 5318	. . . . . . . 8 ⊢ 𝒫 𝐵 ∈ V
13	12	rabex 5277	. . . . . . 7 ⊢ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp} ∈ V
14	10, 1, 13	fvmpt 6942	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp})
15	14	eleq2d 2823	. . . . 5 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp}))
16		oveq2 7369	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝐺 ↾s 𝑠) = (𝐺 ↾s 𝑆))
17	16	eleq1d 2822	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝐺 ↾s 𝑠) ∈ Grp ↔ (𝐺 ↾s 𝑆) ∈ Grp))
18	17	elrab 3635	. . . . . 6 ⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp} ↔ (𝑆 ∈ 𝒫 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))
19	11	elpw2 5272	. . . . . . 7 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵)
20	19	anbi1i 625	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) ↔ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))
21	18, 20	bitri 275	. . . . 5 ⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp} ↔ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))
22	15, 21	bitrdi 287	. . . 4 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)))
23		ibar 528	. . . 4 ⊢ (𝐺 ∈ Grp → ((𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))))
24	22, 23	bitrd 279	. . 3 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))))
25		3anass 1095	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)))
26	24, 25	bitr4di 289	. 2 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)))
27	2, 3, 26	pm5.21nii 378	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {crab 3390   ⊆ wss 3890  𝒫 cpw 4542  ‘cfv 6493  (class class class)co 7361  Basecbs 17173   ↾_s cress 17194  Grpcgrp 18903  SubGrpcsubg 19090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7364  df-subg 19093

This theorem is referenced by:  subgss  19097  subgid  19098  subggrp  19099  subgrcl  19101  issubg2  19111  resgrpisgrp  19117  subsubg  19119  pgrpsubgsymgbi  19377  opprsubg  20326  subrngsubg  20523  subrgsubg  20548  subdrgint  20774  suborng  20847  cphsubrglem  25157  algextdeglem8  33887