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Theorem symgfixelsi 16587
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 16585 . . . 4  |-  ( F  e.  Q  ->  ( F  e.  Q  <->  ( F : N -1-1-onto-> N  /\  ( F `
 K )  =  K ) ) )
4 f1of1 5821 . . . . . . . . . 10  |-  ( F : N -1-1-onto-> N  ->  F : N -1-1-> N )
54ad2antrl 727 . . . . . . . . 9  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  F : N -1-1-> N )
6 difssd 3628 . . . . . . . . 9  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  ( N  \  { K } )  C_  N )
7 f1ores 5836 . . . . . . . . 9  |-  ( ( F : N -1-1-> N  /\  ( N  \  { K } )  C_  N
)  ->  ( F  |`  ( N  \  { K } ) ) : ( N  \  { K } ) -1-1-onto-> ( F " ( N  \  { K }
) ) )
85, 6, 7syl2anc 661 . . . . . . . 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 5280 . . . . . . . . . 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 5829 . . . . . . . . . . . . 13  |-  ( F : N -1-1-onto-> N  ->  F : N -onto-> N )
14 foima 5806 . . . . . . . . . . . . . 14  |-  ( F : N -onto-> N  -> 
( F " N
)  =  N )
1514eqcomd 2465 . . . . . . . . . . . . 13  |-  ( F : N -onto-> N  ->  N  =  ( F " N ) )
1613, 15syl 16 . . . . . . . . . . . 12  |-  ( F : N -1-1-onto-> N  ->  N  =  ( F " N ) )
1716ad2antrl 727 . . . . . . . . . . 11  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  N  =  ( F " N ) )
18 sneq 4042 . . . . . . . . . . . . . 14  |-  ( K  =  ( F `  K )  ->  { K }  =  { ( F `  K ) } )
1918eqcoms 2469 . . . . . . . . . . . . 13  |-  ( ( F `  K )  =  K  ->  { K }  =  { ( F `  K ) } )
2019ad2antll 728 . . . . . . . . . . . 12  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  { K }  =  { ( F `  K ) } )
21 f1ofn 5823 . . . . . . . . . . . . . 14  |-  ( F : N -1-1-onto-> N  ->  F  Fn  N )
2221ad2antrl 727 . . . . . . . . . . . . 13  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  F  Fn  N
)
23 simpl 457 . . . . . . . . . . . . 13  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  K  e.  N
)
24 fnsnfv 5933 . . . . . . . . . . . . 13  |-  ( ( F  Fn  N  /\  K  e.  N )  ->  { ( F `  K ) }  =  ( F " { K } ) )
2522, 23, 24syl2anc 661 . . . . . . . . . . . 12  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  { ( F `
 K ) }  =  ( F " { K } ) )
2620, 25eqtrd 2498 . . . . . . . . . . 11  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  { K }  =  ( F " { K } ) )
2717, 26difeq12d 3619 . . . . . . . . . 10  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  ( N  \  { K } )  =  ( ( F " N )  \  ( F " { K }
) ) )
28 dff1o2 5827 . . . . . . . . . . . . 13  |-  ( F : N -1-1-onto-> N  <->  ( F  Fn  N  /\  Fun  `' F  /\  ran  F  =  N ) )
2928simp2bi 1012 . . . . . . . . . . . 12  |-  ( F : N -1-1-onto-> N  ->  Fun  `' F
)
3029ad2antrl 727 . . . . . . . . . . 11  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  Fun  `' F
)
31 imadif 5669 . . . . . . . . . . 11  |-  ( Fun  `' F  ->  ( F
" ( N  \  { K } ) )  =  ( ( F
" N )  \ 
( F " { K } ) ) )
3230, 31syl 16 . . . . . . . . . 10  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  ( F "
( N  \  { K } ) )  =  ( ( F " N )  \  ( F " { K }
) ) )
3327, 12, 323eqtr4d 2508 . . . . . . . . 9  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  D  =  ( F " ( N 
\  { K }
) ) )
3411, 12, 33f1oeq123d 5819 . . . . . . . 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 232 . . . . . . 7  |-  ( ( K  e.  N  /\  ( F : N -1-1-onto-> N  /\  ( F `  K )  =  K ) )  ->  ( F  |`  D ) : D -1-1-onto-> D
)
3635ancoms 453 . . . . . 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 16586 . . . . . 6  |-  ( F  e.  Q  ->  (
( F  |`  D )  e.  S  <->  ( F  |`  D ) : D -1-1-onto-> D
) )
3936, 38syl5ibr 221 . . . . 5  |-  ( F  e.  Q  ->  (
( ( F : N
-1-1-onto-> N  /\  ( F `  K )  =  K )  /\  K  e.  N )  ->  ( F  |`  D )  e.  S ) )
4039expd 436 . . . 4  |-  ( F  e.  Q  ->  (
( F : N -1-1-onto-> N  /\  ( F `  K
)  =  K )  ->  ( K  e.  N  ->  ( F  |`  D )  e.  S
) ) )
413, 40sylbid 215 . . 3  |-  ( F  e.  Q  ->  ( F  e.  Q  ->  ( K  e.  N  -> 
( F  |`  D )  e.  S ) ) )
4241pm2.43i 47 . 2  |-  ( F  e.  Q  ->  ( K  e.  N  ->  ( F  |`  D )  e.  S ) )
4342impcom 430 1  |-  ( ( K  e.  N  /\  F  e.  Q )  ->  ( F  |`  D )  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {crab 2811    \ cdif 3468    C_ wss 3471   {csn 4032   `'ccnv 5007   ran crn 5009    |` cres 5010   "cima 5011   Fun wfun 5588    Fn wfn 5589   -1-1->wf1 5591   -onto->wfo 5592   -1-1-onto->wf1o 5593   ` cfv 5594   Basecbs 14644   SymGrpcsymg 16529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-plusg 14725  df-tset 14731  df-symg 16530
This theorem is referenced by:  symgfixf  16588  psgnfix1  18761  psgndif  18765  zrhcopsgndif  18766  smadiadetlem3  19297
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