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Theorem symgfixelsi 17076
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
StepHypRef Expression
1 symgfixf.p . . . . 5  |-  P  =  ( Base `  ( SymGrp `
 N ) )
2 symgfixf.q . . . . 5  |-  Q  =  { q  e.  P  |  ( q `  K )  =  K }
31, 2symgfixelq 17074 . . . 4  |-  ( F  e.  Q  ->  ( F  e.  Q  <->  ( F : N -1-1-onto-> N  /\  ( F `
 K )  =  K ) ) )
4 f1of1 5813 . . . . . . . . . 10  |-  ( F : N -1-1-onto-> N  ->  F : N -1-1-> N )
54ad2antrl 734 . . . . . . . . 9  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  F : N -1-1-> N )
6 difssd 3561 . . . . . . . . 9  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  ( N  \  { K } )  C_  N )
7 f1ores 5828 . . . . . . . . 9  |-  ( ( F : N -1-1-> N  /\  ( N  \  { K } )  C_  N
)  ->  ( F  |`  ( N  \  { K } ) ) : ( N  \  { K } ) -1-1-onto-> ( F " ( N  \  { K }
) ) )
85, 6, 7syl2anc 667 . . . . . . . 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 } )
109reseq2i 5102 . . . . . . . . . 10  |-  ( F  |`  D )  =  ( F  |`  ( N  \  { K } ) )
1110a1i 11 . . . . . . . . 9  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  ( F  |`  D )  =  ( F  |`  ( N  \  { K } ) ) )
129a1i 11 . . . . . . . . 9  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  D  =  ( N  \  { K } ) )
13 f1ofo 5821 . . . . . . . . . . . . 13  |-  ( F : N -1-1-onto-> N  ->  F : N -onto-> N )
14 foima 5798 . . . . . . . . . . . . . 14  |-  ( F : N -onto-> N  -> 
( F " N
)  =  N )
1514eqcomd 2457 . . . . . . . . . . . . 13  |-  ( F : N -onto-> N  ->  N  =  ( F " N ) )
1613, 15syl 17 . . . . . . . . . . . 12  |-  ( F : N -1-1-onto-> N  ->  N  =  ( F " N ) )
1716ad2antrl 734 . . . . . . . . . . 11  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  N  =  ( F " N ) )
18 sneq 3978 . . . . . . . . . . . . . 14  |-  ( K  =  ( F `  K )  ->  { K }  =  { ( F `  K ) } )
1918eqcoms 2459 . . . . . . . . . . . . 13  |-  ( ( F `  K )  =  K  ->  { K }  =  { ( F `  K ) } )
2019ad2antll 735 . . . . . . . . . . . 12  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  { K }  =  { ( F `  K ) } )
21 f1ofn 5815 . . . . . . . . . . . . . 14  |-  ( F : N -1-1-onto-> N  ->  F  Fn  N )
2221ad2antrl 734 . . . . . . . . . . . . 13  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  F  Fn  N
)
23 simpl 459 . . . . . . . . . . . . 13  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  K  e.  N
)
24 fnsnfv 5925 . . . . . . . . . . . . 13  |-  ( ( F  Fn  N  /\  K  e.  N )  ->  { ( F `  K ) }  =  ( F " { K } ) )
2522, 23, 24syl2anc 667 . . . . . . . . . . . 12  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  { ( F `
 K ) }  =  ( F " { K } ) )
2620, 25eqtrd 2485 . . . . . . . . . . 11  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  { K }  =  ( F " { K } ) )
2717, 26difeq12d 3552 . . . . . . . . . 10  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  ( N  \  { K } )  =  ( ( F " N )  \  ( F " { K }
) ) )
28 dff1o2 5819 . . . . . . . . . . . . 13  |-  ( F : N -1-1-onto-> N  <->  ( F  Fn  N  /\  Fun  `' F  /\  ran  F  =  N ) )
2928simp2bi 1024 . . . . . . . . . . . 12  |-  ( F : N -1-1-onto-> N  ->  Fun  `' F
)
3029ad2antrl 734 . . . . . . . . . . 11  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  Fun  `' F
)
31 imadif 5658 . . . . . . . . . . 11  |-  ( Fun  `' F  ->  ( F
" ( N  \  { K } ) )  =  ( ( F
" N )  \ 
( F " { K } ) ) )
3230, 31syl 17 . . . . . . . . . 10  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  ( F "
( N  \  { K } ) )  =  ( ( F " N )  \  ( F " { K }
) ) )
3327, 12, 323eqtr4d 2495 . . . . . . . . 9  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  D  =  ( F " ( N 
\  { K }
) ) )
3411, 12, 33f1oeq123d 5811 . . . . . . . 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 }
) ) ) )
358, 34mpbird 236 . . . . . . 7  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  ( F  |`  D ) : D -1-1-onto-> D
)
3635ancoms 455 . . . . . 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 } ) ) )
381, 2, 37, 9symgfixels 17075 . . . . . 6  |-  ( F  e.  Q  ->  (
( F  |`  D )  e.  S  <->  ( F  |`  D ) : D -1-1-onto-> D
) )
3936, 38syl5ibr 225 . . . . 5  |-  ( F  e.  Q  ->  (
( ( F : N
-1-1-onto-> N  /\  ( F `  K )  =  K )  /\  K  e.  N )  ->  ( F  |`  D )  e.  S ) )
4039expd 438 . . . 4  |-  ( F  e.  Q  ->  (
( F : N -1-1-onto-> N  /\  ( F `  K
)  =  K )  ->  ( K  e.  N  ->  ( F  |`  D )  e.  S
) ) )
413, 40sylbid 219 . . 3  |-  ( F  e.  Q  ->  ( F  e.  Q  ->  ( K  e.  N  -> 
( F  |`  D )  e.  S ) ) )
4241pm2.43i 49 . 2  |-  ( F  e.  Q  ->  ( K  e.  N  ->  ( F  |`  D )  e.  S ) )
4342impcom 432 1  |-  ( ( K  e.  N  /\  F  e.  Q )  ->  ( F  |`  D )  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   {crab 2741    \ cdif 3401    C_ wss 3404   {csn 3968   `'ccnv 4833   ran crn 4835    |` cres 4836   "cima 4837   Fun wfun 5576    Fn wfn 5577   -1-1->wf1 5579   -onto->wfo 5580   -1-1-onto->wf1o 5581   ` cfv 5582   Basecbs 15121   SymGrpcsymg 17018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-plusg 15203  df-tset 15209  df-symg 17019
This theorem is referenced by:  symgfixf  17077  psgnfix1  19166  psgndif  19170  zrhcopsgndif  19171  smadiadetlem3  19693
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