Theorem issubg 13759

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 13756	. . 3 |- SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
2	1	mptrcl 5729	. 2 |- ( S e. (SubGrp ` G ) -> G e. Grp )
3		simp1 1023	. 2 |- ( ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) -> G e. Grp )
4		fveq2 5639	. . . . . . . . 9 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
5		issubg.b	. . . . . . . . 9 |- B = ( Base ` G )
6	4, 5	eqtr4di 2282	. . . . . . . 8 |- ( w = G -> ( Base ` w ) = B )
7	6	pweqd 3657	. . . . . . 7 |- ( w = G -> ~P ( Base ` w ) = ~P B )
8		oveq1 6024	. . . . . . . 8 |- ( w = G -> ( w ↾s s ) = ( G ↾s s ) )
9	8	eleq1d 2300	. . . . . . 7 |- ( w = G -> ( ( w ↾s s ) e. Grp <-> ( G ↾s s ) e. Grp ) )
10	7, 9	rabeqbidv 2797	. . . . . 6 |- ( w = G -> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } = { s e. ~P B | ( G ↾s s ) e. Grp } )
11		id 19	. . . . . 6 |- ( G e. Grp -> G e. Grp )
12		basfn 13140	. . . . . . . . . 10 |- Base Fn _V
13		elex 2814	. . . . . . . . . 10 |- ( G e. Grp -> G e. _V )
14		funfvex 5656	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
15	14	funfni 5432	. . . . . . . . . 10 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
16	12, 13, 15	sylancr 414	. . . . . . . . 9 |- ( G e. Grp -> ( Base ` G ) e. _V )
17	5, 16	eqeltrid 2318	. . . . . . . 8 |- ( G e. Grp -> B e. _V )
18	17	pwexd 4271	. . . . . . 7 |- ( G e. Grp -> ~P B e. _V )
19		rabexg 4233	. . . . . . 7 |- ( ~P B e. _V -> { s e. ~P B | ( G ↾s s ) e. Grp } e. _V )
20	18, 19	syl 14	. . . . . 6 |- ( G e. Grp -> { s e. ~P B | ( G ↾s s ) e. Grp } e. _V )
21	1, 10, 11, 20	fvmptd3 5740	. . . . 5 |- ( G e. Grp -> (SubGrp ` G ) = { s e. ~P B | ( G ↾s s ) e. Grp } )
22	21	eleq2d 2301	. . . 4 |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> S e. { s e. ~P B | ( G ↾s s ) e. Grp } ) )
23		oveq2 6025	. . . . . . 7 |- ( s = S -> ( G ↾s s ) = ( G ↾s S ) )
24	23	eleq1d 2300	. . . . . 6 |- ( s = S -> ( ( G ↾s s ) e. Grp <-> ( G ↾s S ) e. Grp ) )
25	24	elrab 2962	. . . . 5 |- ( S e. { s e. ~P B | ( G ↾s s ) e. Grp } <-> ( S e. ~P B /\ ( G ↾s S ) e. Grp ) )
26		elpw2g 4246	. . . . . . 7 |- ( B e. _V -> ( S e. ~P B <-> S C_ B ) )
27	17, 26	syl 14	. . . . . 6 |- ( G e. Grp -> ( S e. ~P B <-> S C_ B ) )
28	27	anbi1d 465	. . . . 5 |- ( G e. Grp -> ( ( S e. ~P B /\ ( G ↾s S ) e. Grp ) <-> ( S C_ B /\ ( G ↾s S ) e. Grp ) ) )
29	25, 28	bitrid 192	. . . 4 |- ( G e. Grp -> ( S e. { s e. ~P B | ( G ↾s s ) e. Grp } <-> ( S C_ B /\ ( G ↾s S ) e. Grp ) ) )
30		ibar 301	. . . 4 |- ( G e. Grp -> ( ( S C_ B /\ ( G ↾s S ) e. Grp ) <-> ( G e. Grp /\ ( S C_ B /\ ( G ↾s S ) e. Grp ) ) ) )
31	22, 29, 30	3bitrd 214	. . 3 |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> ( G e. Grp /\ ( S C_ B /\ ( G ↾s S ) e. Grp ) ) ) )
32		3anass 1008	. . 3 |- ( ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) <-> ( G e. Grp /\ ( S C_ B /\ ( G ↾s S ) e. Grp ) ) )
33	31, 32	bitr4di 198	. 2 |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) ) )
34	2, 3, 33	pm5.21nii 711	1 |- ( S e. (SubGrp ` G ) <-> ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   {crab 2514   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   Fn wfn 5321   ` cfv 5326  (class class class)co 6017   Basecbs 13081   ↾_s cress 13082   Grpcgrp 13582  SubGrpcsubg 13753

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-subg 13756

This theorem is referenced by:  subgss  13760  subgid  13761  subggrp  13763  subgbas  13764  subgrcl  13765  issubg2m  13775  resgrpisgrp  13781  subsubg  13783  opprsubgg  14096  subrngsubg  14217  subrgsubg  14240