Theorem issubg 13246

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 13243	. . 3 ⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
2	1	mptrcl 5641	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3		simp1 999	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) → 𝐺 ∈ Grp)
4		fveq2 5555	. . . . . . . . 9 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
5		issubg.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
6	4, 5	eqtr4di 2244	. . . . . . . 8 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
7	6	pweqd 3607	. . . . . . 7 ⊢ (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵)
8		oveq1 5926	. . . . . . . 8 ⊢ (𝑤 = 𝐺 → (𝑤 ↾s 𝑠) = (𝐺 ↾s 𝑠))
9	8	eleq1d 2262	. . . . . . 7 ⊢ (𝑤 = 𝐺 → ((𝑤 ↾s 𝑠) ∈ Grp ↔ (𝐺 ↾s 𝑠) ∈ Grp))
10	7, 9	rabeqbidv 2755	. . . . . 6 ⊢ (𝑤 = 𝐺 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp} = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp})
11		id 19	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp)
12		basfn 12679	. . . . . . . . . 10 ⊢ Base Fn V
13		elex 2771	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ V)
14		funfvex 5572	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
15	14	funfni 5355	. . . . . . . . . 10 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
16	12, 13, 15	sylancr 414	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → (Base‘𝐺) ∈ V)
17	5, 16	eqeltrid 2280	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ V)
18	17	pwexd 4211	. . . . . . 7 ⊢ (𝐺 ∈ Grp → 𝒫 𝐵 ∈ V)
19		rabexg 4173	. . . . . . 7 ⊢ (𝒫 𝐵 ∈ V → {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp} ∈ V)
20	18, 19	syl 14	. . . . . 6 ⊢ (𝐺 ∈ Grp → {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp} ∈ V)
21	1, 10, 11, 20	fvmptd3 5652	. . . . 5 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp})
22	21	eleq2d 2263	. . . 4 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp}))
23		oveq2 5927	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝐺 ↾s 𝑠) = (𝐺 ↾s 𝑆))
24	23	eleq1d 2262	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝐺 ↾s 𝑠) ∈ Grp ↔ (𝐺 ↾s 𝑆) ∈ Grp))
25	24	elrab 2917	. . . . 5 ⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp} ↔ (𝑆 ∈ 𝒫 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))
26		elpw2g 4186	. . . . . . 7 ⊢ (𝐵 ∈ V → (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵))
27	17, 26	syl 14	. . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵))
28	27	anbi1d 465	. . . . 5 ⊢ (𝐺 ∈ Grp → ((𝑆 ∈ 𝒫 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) ↔ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)))
29	25, 28	bitrid 192	. . . 4 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp} ↔ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)))
30		ibar 301	. . . 4 ⊢ (𝐺 ∈ Grp → ((𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))))
31	22, 29, 30	3bitrd 214	. . 3 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))))
32		3anass 984	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)))
33	31, 32	bitr4di 198	. 2 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)))
34	2, 3, 33	pm5.21nii 705	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  {crab 2476  Vcvv 2760   ⊆ wss 3154  𝒫 cpw 3602   Fn wfn 5250  ‘cfv 5255  (class class class)co 5919  Basecbs 12621   ↾_s cress 12622  Grpcgrp 13075  SubGrpcsubg 13240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922  df-inn 8985  df-ndx 12624  df-slot 12625  df-base 12627  df-subg 13243

This theorem is referenced by:  subgss  13247  subgid  13248  subggrp  13250  subgbas  13251  subgrcl  13252  issubg2m  13262  resgrpisgrp  13268  subsubg  13270  opprsubgg  13583  subrngsubg  13703  subrgsubg  13726