Theorem symgfixelsi 17855

Description: The restriction of a permutation fixing an element to the set with this element removed is an element of the restricted symmetric group. (Contributed by AV, 4-Jan-2019.)

Hypotheses

Ref	Expression
symgfixf.p	|- P = ( Base ` ( SymGrp ` N ) )
symgfixf.q	|- Q = { q e. P | ( q ` K ) = K }
symgfixf.s	|- S = ( Base ` ( SymGrp ` ( N \ { K } ) ) )
symgfixf.d	|- D = ( N \ { K } )

Assertion

Ref	Expression
symgfixelsi	|- ( ( K e. N /\ F e. Q ) -> ( F |` D ) e. S )

Distinct variable groups:   K, q   P, q

Allowed substitution hints:   D( q)   Q( q)   S( q)   F( q)   N( q)

Proof of Theorem symgfixelsi

Step	Hyp	Ref	Expression
1		symgfixf.p	. . . . 5 |- P = ( Base ` ( SymGrp ` N ) )
2		symgfixf.q	. . . . 5 |- Q = { q e. P | ( q ` K ) = K }
3	1, 2	symgfixelq 17853	. . . 4 |- ( F e. Q -> ( F e. Q <-> ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) )
4		f1of1 6136	. . . . . . . . . 10 |- ( F : N -1-1-onto-> N -> F : N -1-1-> N )
5	4	ad2antrl 764	. . . . . . . . 9 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> F : N -1-1-> N )
6		difssd 3738	. . . . . . . . 9 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> ( N \ { K } ) C_ N )
7		f1ores 6151	. . . . . . . . 9 |- ( ( F : N -1-1-> N /\ ( N \ { K } ) C_ N ) -> ( F |` ( N \ { K } ) ) : ( N \ { K } ) -1-1-onto-> ( F " ( N \ { K } ) ) )
8	5, 6, 7	syl2anc 693	. . . . . . . 8 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> ( F |` ( N \ { K } ) ) : ( N \ { K } ) -1-1-onto-> ( F " ( N \ { K } ) ) )
9		symgfixf.d	. . . . . . . . . . 11 |- D = ( N \ { K } )
10	9	reseq2i 5393	. . . . . . . . . 10 |- ( F |` D ) = ( F |` ( N \ { K } ) )
11	10	a1i 11	. . . . . . . . 9 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> ( F |` D ) = ( F |` ( N \ { K } ) ) )
12	9	a1i 11	. . . . . . . . 9 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> D = ( N \ { K } ) )
13		f1ofo 6144	. . . . . . . . . . . . 13 |- ( F : N -1-1-onto-> N -> F : N -onto-> N )
14		foima 6120	. . . . . . . . . . . . . 14 |- ( F : N -onto-> N -> ( F " N ) = N )
15	14	eqcomd 2628	. . . . . . . . . . . . 13 |- ( F : N -onto-> N -> N = ( F " N ) )
16	13, 15	syl 17	. . . . . . . . . . . 12 |- ( F : N -1-1-onto-> N -> N = ( F " N ) )
17	16	ad2antrl 764	. . . . . . . . . . 11 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> N = ( F " N ) )
18		sneq 4187	. . . . . . . . . . . . . 14 |- ( K = ( F ` K ) -> { K } = { ( F ` K ) } )
19	18	eqcoms 2630	. . . . . . . . . . . . 13 |- ( ( F ` K ) = K -> { K } = { ( F ` K ) } )
20	19	ad2antll 765	. . . . . . . . . . . 12 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> { K } = { ( F ` K ) } )
21		f1ofn 6138	. . . . . . . . . . . . . 14 |- ( F : N -1-1-onto-> N -> F Fn N )
22	21	ad2antrl 764	. . . . . . . . . . . . 13 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> F Fn N )
23		simpl 473	. . . . . . . . . . . . 13 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> K e. N )
24		fnsnfv 6258	. . . . . . . . . . . . 13 |- ( ( F Fn N /\ K e. N ) -> { ( F ` K ) } = ( F " { K } ) )
25	22, 23, 24	syl2anc 693	. . . . . . . . . . . 12 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> { ( F ` K ) } = ( F " { K } ) )
26	20, 25	eqtrd 2656	. . . . . . . . . . 11 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> { K } = ( F " { K } ) )
27	17, 26	difeq12d 3729	. . . . . . . . . 10 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> ( N \ { K } ) = ( ( F " N ) \ ( F " { K } ) ) )
28		dff1o2 6142	. . . . . . . . . . . . 13 |- ( F : N -1-1-onto-> N <-> ( F Fn N /\ Fun `' F /\ ran F = N ) )
29	28	simp2bi 1077	. . . . . . . . . . . 12 |- ( F : N -1-1-onto-> N -> Fun `' F )
30	29	ad2antrl 764	. . . . . . . . . . 11 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> Fun `' F )
31		imadif 5973	. . . . . . . . . . 11 |- ( Fun `' F -> ( F " ( N \ { K } ) ) = ( ( F " N ) \ ( F " { K } ) ) )
32	30, 31	syl 17	. . . . . . . . . 10 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> ( F " ( N \ { K } ) ) = ( ( F " N ) \ ( F " { K } ) ) )
33	27, 12, 32	3eqtr4d 2666	. . . . . . . . 9 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> D = ( F " ( N \ { K } ) ) )
34	11, 12, 33	f1oeq123d 6133	. . . . . . . 8 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> ( ( F |` D ) : D -1-1-onto-> D <-> ( F |` ( N \ { K } ) ) : ( N \ { K } ) -1-1-onto-> ( F " ( N \ { K } ) ) ) )
35	8, 34	mpbird 247	. . . . . . 7 |- ( ( K e. N /\ ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) ) -> ( F |` D ) : D -1-1-onto-> D )
36	35	ancoms 469	. . . . . 6 |- ( ( ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) /\ K e. N ) -> ( F |` D ) : D -1-1-onto-> D )
37		symgfixf.s	. . . . . . 7 |- S = ( Base ` ( SymGrp ` ( N \ { K } ) ) )
38	1, 2, 37, 9	symgfixels 17854	. . . . . 6 |- ( F e. Q -> ( ( F |` D ) e. S <-> ( F |` D ) : D -1-1-onto-> D ) )
39	36, 38	syl5ibr 236	. . . . 5 |- ( F e. Q -> ( ( ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) /\ K e. N ) -> ( F |` D ) e. S ) )
40	39	expd 452	. . . 4 |- ( F e. Q -> ( ( F : N -1-1-onto-> N /\ ( F ` K ) = K ) -> ( K e. N -> ( F |` D ) e. S ) ) )
41	3, 40	sylbid 230	. . 3 |- ( F e. Q -> ( F e. Q -> ( K e. N -> ( F |` D ) e. S ) ) )
42	41	pm2.43i 52	. 2 |- ( F e. Q -> ( K e. N -> ( F |` D ) e. S ) )
43	42	impcom 446	1 |- ( ( K e. N /\ F e. Q ) -> ( F |` D ) e. S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   C_ wss 3574   {csn 4177   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   Basecbs 15857   SymGrpcsymg 17797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-tset 15960  df-symg 17798

This theorem is referenced by:  symgfixf  17856  psgnfix1  19944  psgndif  19948  zrhcopsgndif  19949  smadiadetlem3  20474