Theorem conjsubgen 13864

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 13765	. . . . . . 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 2231	. . . . . . . 8 |- ( x e. X |-> ( ( A .+ x ) .- A ) ) = ( x e. X |-> ( ( A .+ x ) .- A ) )
6	2, 3, 4, 5	conjghm 13862	. . . . . . 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 5582	. . . . . 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 13760	. . . . . 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 5551	. . . . 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 5064	. . . . . 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 2282	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ( ( x e. X |-> ( ( A .+ x ) .- A ) ) |` S ) = F )
17		f1eq1 5537	. . . . 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 5594	. . 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 6929	. 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 1397   e. wcel 2202   C_ wss 3200   class class class wbr 4088   |-> cmpt 4150   ran crn 4726   |` cres 4727   -1-1->wf1 5323   -1-1-onto->wf1o 5325   ` cfv 5326  (class class class)co 6017   ~~ cen 6906   Basecbs 13081   +g cplusg 13159   Grpcgrp 13582   -gcsg 13584  SubGrpcsubg 13753   GrpHom cghm 13826

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-en 6909  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-sbg 13587  df-subg 13756  df-ghm 13827

This theorem is referenced by: (None)