Theorem conjsubgen 13830

Description: A conjugated subgroup is equinumerous to the original subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
conjghm.x	|- X = ( Base ` G )
conjghm.p	|- .+ = ( +g ` G )
conjghm.m	|- .- = ( -g ` G )
conjsubg.f	|- F = ( x e. S |-> ( ( A .+ x ) .- A ) )

Assertion

Ref	Expression
conjsubgen	|- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> S ~~ ran F )

Distinct variable groups:   x, .-   x, .+   x, A   x, G   x, S   x, X

Allowed substitution hint:   F( x)

Proof of Theorem conjsubgen

Step	Hyp	Ref	Expression
1		subgrcl 13731	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> G e. Grp )
2		conjghm.x	. . . . . . . 8 |- X = ( Base ` G )
3		conjghm.p	. . . . . . . 8 |- .+ = ( +g ` G )
4		conjghm.m	. . . . . . . 8 |- .- = ( -g ` G )
5		eqid 2229	. . . . . . . 8 |- ( x e. X |-> ( ( A .+ x ) .- A ) ) = ( x e. X |-> ( ( A .+ x ) .- A ) )
6	2, 3, 4, 5	conjghm 13828	. . . . . . 7 |- ( ( G e. Grp /\ A e. X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) e. ( G GrpHom G ) /\ ( x e. X |-> ( ( A .+ x ) .- A ) ) : X -1-1-onto-> X ) )
7	1, 6	sylan 283	. . . . . 6 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) e. ( G GrpHom G ) /\ ( x e. X |-> ( ( A .+ x ) .- A ) ) : X -1-1-onto-> X ) )
8		f1of1 5573	. . . . . 6 |- ( ( x e. X |-> ( ( A .+ x ) .- A ) ) : X -1-1-onto-> X -> ( x e. X |-> ( ( A .+ x ) .- A ) ) : X -1-1-> X )
9	7, 8	simpl2im 386	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( x e. X |-> ( ( A .+ x ) .- A ) ) : X -1-1-> X )
10	2	subgss 13726	. . . . . 6 |- ( S e. (SubGrp ` G ) -> S C_ X )
11	10	adantr 276	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> S C_ X )
12		f1ssres 5542	. . . . 5 |- ( ( ( x e. X |-> ( ( A .+ x ) .- A ) ) : X -1-1-> X /\ S C_ X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) : S -1-1-> X )
13	9, 11, 12	syl2anc 411	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) : S -1-1-> X )
14	11	resmptd 5056	. . . . . 6 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = ( x e. S |-> ( ( A .+ x ) .- A ) ) )
15		conjsubg.f	. . . . . 6 |- F = ( x e. S |-> ( ( A .+ x ) .- A ) )
16	14, 15	eqtr4di 2280	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = F )
17		f1eq1 5528	. . . . 5 |- ( ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = F -> ( ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) : S -1-1-> X <-> F : S -1-1-> X ) )
18	16, 17	syl 14	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) : S -1-1-> X <-> F : S -1-1-> X ) )
19	13, 18	mpbid 147	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> F : S -1-1-> X )
20		f1f1orn 5585	. . 3 |- ( F : S -1-1-> X -> F : S -1-1-onto-> ran F )
21	19, 20	syl 14	. 2 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> F : S -1-1-onto-> ran F )
22		f1oeng 6916	. 2 |- ( ( S e. (SubGrp ` G ) /\ F : S -1-1-onto-> ran F ) -> S ~~ ran F )
23	21, 22	syldan 282	1 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> S ~~ ran F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   C_ wss 3197   class class class wbr 4083   |-> cmpt 4145   ran crn 4720   |` cres 4721   -1-1->wf1 5315   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6007   ~~ cen 6893   Basecbs 13047   +g cplusg 13125   Grpcgrp 13548   -gcsg 13550  SubGrpcsubg 13719   GrpHom cghm 13792

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1re 8104  ax-addrcl 8107

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-en 6896  df-inn 9122  df-2 9180  df-ndx 13050  df-slot 13051  df-base 13053  df-plusg 13138  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-grp 13551  df-minusg 13552  df-sbg 13553  df-subg 13722  df-ghm 13793

This theorem is referenced by: (None)