Theorem issubg 13065

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 13062	. . 3 |- SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
2	1	mptrcl 5611	. 2 |- ( S e. (SubGrp ` G ) -> G e. Grp )
3		simp1 998	. 2 |- ( ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) -> G e. Grp )
4		fveq2 5527	. . . . . . . . 9 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
5		issubg.b	. . . . . . . . 9 |- B = ( Base ` G )
6	4, 5	eqtr4di 2238	. . . . . . . 8 |- ( w = G -> ( Base ` w ) = B )
7	6	pweqd 3592	. . . . . . 7 |- ( w = G -> ~P ( Base ` w ) = ~P B )
8		oveq1 5895	. . . . . . . 8 |- ( w = G -> ( w ↾s s ) = ( G ↾s s ) )
9	8	eleq1d 2256	. . . . . . 7 |- ( w = G -> ( ( w ↾s s ) e. Grp <-> ( G ↾s s ) e. Grp ) )
10	7, 9	rabeqbidv 2744	. . . . . 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 12534	. . . . . . . . . 10 |- Base Fn _V
13		elex 2760	. . . . . . . . . 10 |- ( G e. Grp -> G e. _V )
14		funfvex 5544	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
15	14	funfni 5328	. . . . . . . . . 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 2274	. . . . . . . 8 |- ( G e. Grp -> B e. _V )
18	17	pwexd 4193	. . . . . . 7 |- ( G e. Grp -> ~P B e. _V )
19		rabexg 4158	. . . . . . 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 5622	. . . . 5 |- ( G e. Grp -> (SubGrp ` G ) = { s e. ~P B | ( G ↾s s ) e. Grp } )
22	21	eleq2d 2257	. . . 4 |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> S e. { s e. ~P B | ( G ↾s s ) e. Grp } ) )
23		oveq2 5896	. . . . . . 7 |- ( s = S -> ( G ↾s s ) = ( G ↾s S ) )
24	23	eleq1d 2256	. . . . . 6 |- ( s = S -> ( ( G ↾s s ) e. Grp <-> ( G ↾s S ) e. Grp ) )
25	24	elrab 2905	. . . . 5 |- ( S e. { s e. ~P B | ( G ↾s s ) e. Grp } <-> ( S e. ~P B /\ ( G ↾s S ) e. Grp ) )
26		elpw2g 4168	. . . . . . 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 983	. . 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 705	1 |- ( S e. (SubGrp ` G ) <-> ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 979   = wceq 1363   e. wcel 2158   {crab 2469   _Vcvv 2749   C_ wss 3141   ~Pcpw 3587   Fn wfn 5223   ` cfv 5228  (class class class)co 5888   Basecbs 12476   ↾_s cress 12477   Grpcgrp 12899  SubGrpcsubg 13059

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7916  ax-resscn 7917  ax-1re 7919  ax-addrcl 7922

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-ov 5891  df-inn 8934  df-ndx 12479  df-slot 12480  df-base 12482  df-subg 13062

This theorem is referenced by:  subgss  13066  subgid  13067  subggrp  13069  subgbas  13070  subgrcl  13071  issubg2m  13081  resgrpisgrp  13087  subsubg  13089  opprsubgg  13332  subrngsubg  13424  subrgsubg  13447