Theorem issubg 13542

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 13539	. . 3 |- SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
2	1	mptrcl 5664	. 2 |- ( S e. (SubGrp ` G ) -> G e. Grp )
3		simp1 1000	. 2 |- ( ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) -> G e. Grp )
4		fveq2 5578	. . . . . . . . 9 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
5		issubg.b	. . . . . . . . 9 |- B = ( Base ` G )
6	4, 5	eqtr4di 2256	. . . . . . . 8 |- ( w = G -> ( Base ` w ) = B )
7	6	pweqd 3621	. . . . . . 7 |- ( w = G -> ~P ( Base ` w ) = ~P B )
8		oveq1 5953	. . . . . . . 8 |- ( w = G -> ( w ↾s s ) = ( G ↾s s ) )
9	8	eleq1d 2274	. . . . . . 7 |- ( w = G -> ( ( w ↾s s ) e. Grp <-> ( G ↾s s ) e. Grp ) )
10	7, 9	rabeqbidv 2767	. . . . . 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 12923	. . . . . . . . . 10 |- Base Fn _V
13		elex 2783	. . . . . . . . . 10 |- ( G e. Grp -> G e. _V )
14		funfvex 5595	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
15	14	funfni 5377	. . . . . . . . . 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 2292	. . . . . . . 8 |- ( G e. Grp -> B e. _V )
18	17	pwexd 4226	. . . . . . 7 |- ( G e. Grp -> ~P B e. _V )
19		rabexg 4188	. . . . . . 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 5675	. . . . 5 |- ( G e. Grp -> (SubGrp ` G ) = { s e. ~P B | ( G ↾s s ) e. Grp } )
22	21	eleq2d 2275	. . . 4 |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> S e. { s e. ~P B | ( G ↾s s ) e. Grp } ) )
23		oveq2 5954	. . . . . . 7 |- ( s = S -> ( G ↾s s ) = ( G ↾s S ) )
24	23	eleq1d 2274	. . . . . 6 |- ( s = S -> ( ( G ↾s s ) e. Grp <-> ( G ↾s S ) e. Grp ) )
25	24	elrab 2929	. . . . 5 |- ( S e. { s e. ~P B | ( G ↾s s ) e. Grp } <-> ( S e. ~P B /\ ( G ↾s S ) e. Grp ) )
26		elpw2g 4201	. . . . . . 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 985	. . 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 706	1 |- ( S e. (SubGrp ` G ) <-> ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   {crab 2488   _Vcvv 2772   C_ wss 3166   ~Pcpw 3616   Fn wfn 5267   ` cfv 5272  (class class class)co 5946   Basecbs 12865   ↾_s cress 12866   Grpcgrp 13365  SubGrpcsubg 13536

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-cnex 8018  ax-resscn 8019  ax-1re 8021  ax-addrcl 8024

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280  df-ov 5949  df-inn 9039  df-ndx 12868  df-slot 12869  df-base 12871  df-subg 13539

This theorem is referenced by:  subgss  13543  subgid  13544  subggrp  13546  subgbas  13547  subgrcl  13548  issubg2m  13558  resgrpisgrp  13564  subsubg  13566  opprsubgg  13879  subrngsubg  13999  subrgsubg  14022