Theorem conjsubgen 13855

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 13756	. . . . . . 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 13853	. . . . . . 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 5579	. . . . . 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 13751	. . . . . 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 5548	. . . . 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 5062	. . . . . 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 5534	. . . . 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 5591	. . 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 6925	. 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 3198   class class class wbr 4086   |-> cmpt 4148   ran crn 4724   |` cres 4725   -1-1->wf1 5321   -1-1-onto->wf1o 5323   ` cfv 5324  (class class class)co 6013   ~~ cen 6902   Basecbs 13072   +g cplusg 13150   Grpcgrp 13573   -gcsg 13575  SubGrpcsubg 13744   GrpHom cghm 13817

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 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-en 6905  df-inn 9134  df-2 9192  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-grp 13576  df-minusg 13577  df-sbg 13578  df-subg 13747  df-ghm 13818

This theorem is referenced by: (None)