Theorem issubg 13926

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 13923	. . 3 |- SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
2	1	mptrcl 5765	. 2 |- ( S e. (SubGrp ` G ) -> G e. Grp )
3		simp1 1024	. 2 |- ( ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) -> G e. Grp )
4		fveq2 5675	. . . . . . . . 9 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
5		issubg.b	. . . . . . . . 9 |- B = ( Base ` G )
6	4, 5	eqtr4di 2285	. . . . . . . 8 |- ( w = G -> ( Base ` w ) = B )
7	6	pweqd 3679	. . . . . . 7 |- ( w = G -> ~P ( Base ` w ) = ~P B )
8		oveq1 6065	. . . . . . . 8 |- ( w = G -> ( w ↾s s ) = ( G ↾s s ) )
9	8	eleq1d 2303	. . . . . . 7 |- ( w = G -> ( ( w ↾s s ) e. Grp <-> ( G ↾s s ) e. Grp ) )
10	7, 9	rabeqbidv 2810	. . . . . 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 13355	. . . . . . . . . 10 |- Base Fn _V
13		elex 2827	. . . . . . . . . 10 |- ( G e. Grp -> G e. _V )
14		funfvex 5692	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
15	14	funfni 5463	. . . . . . . . . 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 2321	. . . . . . . 8 |- ( G e. Grp -> B e. _V )
18	17	pwexd 4299	. . . . . . 7 |- ( G e. Grp -> ~P B e. _V )
19		rabexg 4260	. . . . . . 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 5776	. . . . 5 |- ( G e. Grp -> (SubGrp ` G ) = { s e. ~P B | ( G ↾s s ) e. Grp } )
22	21	eleq2d 2304	. . . 4 |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> S e. { s e. ~P B | ( G ↾s s ) e. Grp } ) )
23		oveq2 6066	. . . . . . 7 |- ( s = S -> ( G ↾s s ) = ( G ↾s S ) )
24	23	eleq1d 2303	. . . . . 6 |- ( s = S -> ( ( G ↾s s ) e. Grp <-> ( G ↾s S ) e. Grp ) )
25	24	elrab 2976	. . . . 5 |- ( S e. { s e. ~P B | ( G ↾s s ) e. Grp } <-> ( S e. ~P B /\ ( G ↾s S ) e. Grp ) )
26		elpw2g 4273	. . . . . . 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 1009	. . 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 712	1 |- ( S e. (SubGrp ` G ) <-> ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   {crab 2526   _Vcvv 2815   C_ wss 3214   ~Pcpw 3674   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   Basecbs 13296   ↾_s cress 13297   Grpcgrp 13755  SubGrpcsubg 13920

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302  df-subg 13923

This theorem is referenced by:  subgss  13927  subgid  13928  subggrp  13930  subgbas  13931  subgrcl  13932  issubg2m  13942  resgrpisgrp  13948  subsubg  13950  opprsubgg  14328  subrngsubg  14450  subrgsubg  14473