Theorem issubg 13038

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 13035	. . 3 |- SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
2	1	mptrcl 5600	. 2 |- ( S e. (SubGrp ` G ) -> G e. Grp )
3		simp1 997	. 2 |- ( ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) -> G e. Grp )
4		fveq2 5517	. . . . . . . . 9 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
5		issubg.b	. . . . . . . . 9 |- B = ( Base ` G )
6	4, 5	eqtr4di 2228	. . . . . . . 8 |- ( w = G -> ( Base ` w ) = B )
7	6	pweqd 3582	. . . . . . 7 |- ( w = G -> ~P ( Base ` w ) = ~P B )
8		oveq1 5884	. . . . . . . 8 |- ( w = G -> ( w ↾s s ) = ( G ↾s s ) )
9	8	eleq1d 2246	. . . . . . 7 |- ( w = G -> ( ( w ↾s s ) e. Grp <-> ( G ↾s s ) e. Grp ) )
10	7, 9	rabeqbidv 2734	. . . . . 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 12522	. . . . . . . . . 10 |- Base Fn _V
13		elex 2750	. . . . . . . . . 10 |- ( G e. Grp -> G e. _V )
14		funfvex 5534	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
15	14	funfni 5318	. . . . . . . . . 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 2264	. . . . . . . 8 |- ( G e. Grp -> B e. _V )
18	17	pwexd 4183	. . . . . . 7 |- ( G e. Grp -> ~P B e. _V )
19		rabexg 4148	. . . . . . 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 5611	. . . . 5 |- ( G e. Grp -> (SubGrp ` G ) = { s e. ~P B | ( G ↾s s ) e. Grp } )
22	21	eleq2d 2247	. . . 4 |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> S e. { s e. ~P B | ( G ↾s s ) e. Grp } ) )
23		oveq2 5885	. . . . . . 7 |- ( s = S -> ( G ↾s s ) = ( G ↾s S ) )
24	23	eleq1d 2246	. . . . . 6 |- ( s = S -> ( ( G ↾s s ) e. Grp <-> ( G ↾s S ) e. Grp ) )
25	24	elrab 2895	. . . . 5 |- ( S e. { s e. ~P B | ( G ↾s s ) e. Grp } <-> ( S e. ~P B /\ ( G ↾s S ) e. Grp ) )
26		elpw2g 4158	. . . . . . 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 982	. . 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 704	1 |- ( S e. (SubGrp ` G ) <-> ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   {crab 2459   _Vcvv 2739   C_ wss 3131   ~Pcpw 3577   Fn wfn 5213   ` cfv 5218  (class class class)co 5877   Basecbs 12464   ↾_s cress 12465   Grpcgrp 12882  SubGrpcsubg 13032

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ov 5880  df-inn 8922  df-ndx 12467  df-slot 12468  df-base 12470  df-subg 13035

This theorem is referenced by:  subgss  13039  subgid  13040  subggrp  13042  subgbas  13043  subgrcl  13044  issubg2m  13054  resgrpisgrp  13060  subsubg  13062  subrgsubg  13353