Theorem issubg 17594

Description: The subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)

Hypothesis

Ref	Expression
issubg.b	|- B = ( Base ` G )

Assertion

Ref	Expression
issubg	|- ( S e. (SubGrp ` G ) <-> ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. 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 17591	. . . 4 |- SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
2	1	dmmptss 5631	. . 3 |- dom SubGrp C_ Grp
3		elfvdm 6220	. . 3 |- ( S e. (SubGrp ` G ) -> G e. dom SubGrp )
4	2, 3	sseldi 3601	. 2 |- ( S e. (SubGrp ` G ) -> G e. Grp )
5		simp1 1061	. 2 |- ( ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) -> G e. Grp )
6		fveq2 6191	. . . . . . . . . 10 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
7		issubg.b	. . . . . . . . . 10 |- B = ( Base ` G )
8	6, 7	syl6eqr 2674	. . . . . . . . 9 |- ( w = G -> ( Base ` w ) = B )
9	8	pweqd 4163	. . . . . . . 8 |- ( w = G -> ~P ( Base ` w ) = ~P B )
10		oveq1 6657	. . . . . . . . 9 |- ( w = G -> ( w ↾s s ) = ( G ↾s s ) )
11	10	eleq1d 2686	. . . . . . . 8 |- ( w = G -> ( ( w ↾s s ) e. Grp <-> ( G ↾s s ) e. Grp ) )
12	9, 11	rabeqbidv 3195	. . . . . . 7 |- ( w = G -> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } = { s e. ~P B | ( G ↾s s ) e. Grp } )
13		fvex 6201	. . . . . . . . . 10 |- ( Base ` G ) e. _V
14	7, 13	eqeltri 2697	. . . . . . . . 9 |- B e. _V
15	14	pwex 4848	. . . . . . . 8 |- ~P B e. _V
16	15	rabex 4813	. . . . . . 7 |- { s e. ~P B | ( G ↾s s ) e. Grp } e. _V
17	12, 1, 16	fvmpt 6282	. . . . . 6 |- ( G e. Grp -> (SubGrp ` G ) = { s e. ~P B | ( G ↾s s ) e. Grp } )
18	17	eleq2d 2687	. . . . 5 |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> S e. { s e. ~P B | ( G ↾s s ) e. Grp } ) )
19		oveq2 6658	. . . . . . . 8 |- ( s = S -> ( G ↾s s ) = ( G ↾s S ) )
20	19	eleq1d 2686	. . . . . . 7 |- ( s = S -> ( ( G ↾s s ) e. Grp <-> ( G ↾s S ) e. Grp ) )
21	20	elrab 3363	. . . . . 6 |- ( S e. { s e. ~P B | ( G ↾s s ) e. Grp } <-> ( S e. ~P B /\ ( G ↾s S ) e. Grp ) )
22	14	elpw2 4828	. . . . . . 7 |- ( S e. ~P B <-> S C_ B )
23	22	anbi1i 731	. . . . . 6 |- ( ( S e. ~P B /\ ( G ↾s S ) e. Grp ) <-> ( S C_ B /\ ( G ↾s S ) e. Grp ) )
24	21, 23	bitri 264	. . . . 5 |- ( S e. { s e. ~P B | ( G ↾s s ) e. Grp } <-> ( S C_ B /\ ( G ↾s S ) e. Grp ) )
25	18, 24	syl6bb 276	. . . 4 |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> ( S C_ B /\ ( G ↾s S ) e. Grp ) ) )
26		ibar 525	. . . 4 |- ( G e. Grp -> ( ( S C_ B /\ ( G ↾s S ) e. Grp ) <-> ( G e. Grp /\ ( S C_ B /\ ( G ↾s S ) e. Grp ) ) ) )
27	25, 26	bitrd 268	. . 3 |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> ( G e. Grp /\ ( S C_ B /\ ( G ↾s S ) e. Grp ) ) ) )
28		3anass 1042	. . 3 |- ( ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) <-> ( G e. Grp /\ ( S C_ B /\ ( G ↾s S ) e. Grp ) ) )
29	27, 28	syl6bbr 278	. 2 |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) ) )
30	4, 5, 29	pm5.21nii 368	1 |- ( S e. (SubGrp ` G ) <-> ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   dom cdm 5114   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   Grpcgrp 17422  SubGrpcsubg 17588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-subg 17591

This theorem is referenced by:  subgss  17595  subgid  17596  subggrp  17597  subgrcl  17599  issubg2  17609  resgrpisgrp  17615  subsubg  17617  pgrpsubgsymgbi  17827  opprsubg  18636  subrgsubg  18786  cphsubrglem  22977  suborng  29815