Theorem conjsubgen 13484

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 13385	. . . . . . 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 2196	. . . . . . . 8 |- ( x e. X |-> ( ( A .+ x ) .- A ) ) = ( x e. X |-> ( ( A .+ x ) .- A ) )
6	2, 3, 4, 5	conjghm 13482	. . . . . . 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 5506	. . . . . 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 13380	. . . . . 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 5475	. . . . 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 4998	. . . . . 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 2247	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = F )
17		f1eq1 5461	. . . . 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 5518	. . 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 6825	. 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 1364   e. wcel 2167   C_ wss 3157   class class class wbr 4034   |-> cmpt 4095   ran crn 4665   |` cres 4666   -1-1->wf1 5256   -1-1-onto->wf1o 5258   ` cfv 5259  (class class class)co 5925   ~~ cen 6806   Basecbs 12703   +g cplusg 12780   Grpcgrp 13202   -gcsg 13204  SubGrpcsubg 13373   GrpHom cghm 13446

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-en 6809  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-sbg 13207  df-subg 13376  df-ghm 13447

This theorem is referenced by: (None)