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Theorem symgfixfo 16591
Description: The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is an onto function. (Contributed by AV, 7-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.h  |-  H  =  ( q  e.  Q  |->  ( q  |`  ( N  \  { K }
) ) )
Assertion
Ref Expression
symgfixfo  |-  ( ( N  e.  V  /\  K  e.  N )  ->  H : Q -onto-> S
)
Distinct variable groups:    K, q    P, q    N, q    Q, q    S, q
Allowed substitution hints:    H( q)    V( q)

Proof of Theorem symgfixfo
Dummy variables  p  i  s  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 symgfixf.p . . . 4  |-  P  =  ( Base `  ( SymGrp `
 N ) )
2 symgfixf.q . . . 4  |-  Q  =  { q  e.  P  |  ( q `  K )  =  K }
3 symgfixf.s . . . 4  |-  S  =  ( Base `  ( SymGrp `
 ( N  \  { K } ) ) )
4 symgfixf.h . . . 4  |-  H  =  ( q  e.  Q  |->  ( q  |`  ( N  \  { K }
) ) )
51, 2, 3, 4symgfixf 16588 . . 3  |-  ( K  e.  N  ->  H : Q --> S )
65adantl 466 . 2  |-  ( ( N  e.  V  /\  K  e.  N )  ->  H : Q --> S )
7 eqeq1 2461 . . . . . . . . . 10  |-  ( i  =  j  ->  (
i  =  K  <->  j  =  K ) )
8 fveq2 5872 . . . . . . . . . 10  |-  ( i  =  j  ->  (
s `  i )  =  ( s `  j ) )
97, 8ifbieq2d 3969 . . . . . . . . 9  |-  ( i  =  j  ->  if ( i  =  K ,  K ,  ( s `  i ) )  =  if ( j  =  K ,  K ,  ( s `  j ) ) )
109cbvmptv 4548 . . . . . . . 8  |-  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  =  ( j  e.  N  |->  if ( j  =  K ,  K ,  ( s `  j ) ) )
111, 2, 3, 4, 10symgfixfolem1 16590 . . . . . . 7  |-  ( ( N  e.  V  /\  K  e.  N  /\  s  e.  S )  ->  ( i  e.  N  |->  if ( i  =  K ,  K , 
( s `  i
) ) )  e.  Q )
12113expa 1196 . . . . . 6  |-  ( ( ( N  e.  V  /\  K  e.  N
)  /\  s  e.  S )  ->  (
i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  e.  Q
)
13 simpr 461 . . . . . . . . . . . . 13  |-  ( ( N  e.  V  /\  K  e.  N )  ->  K  e.  N )
1413anim1i 568 . . . . . . . . . . . 12  |-  ( ( ( N  e.  V  /\  K  e.  N
)  /\  s  e.  S )  ->  ( K  e.  N  /\  s  e.  S )
)
1514adantl 466 . . . . . . . . . . 11  |-  ( ( p  =  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  /\  ( ( N  e.  V  /\  K  e.  N )  /\  s  e.  S ) )  -> 
( K  e.  N  /\  s  e.  S
) )
16 eqid 2457 . . . . . . . . . . . 12  |-  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  =  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )
173, 16symgextres 16577 . . . . . . . . . . 11  |-  ( ( K  e.  N  /\  s  e.  S )  ->  ( ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  |`  ( N  \  { K } ) )  =  s )
1815, 17syl 16 . . . . . . . . . 10  |-  ( ( p  =  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  /\  ( ( N  e.  V  /\  K  e.  N )  /\  s  e.  S ) )  -> 
( ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  |`  ( N  \  { K } ) )  =  s )
1918eqcomd 2465 . . . . . . . . 9  |-  ( ( p  =  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  /\  ( ( N  e.  V  /\  K  e.  N )  /\  s  e.  S ) )  -> 
s  =  ( ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  |`  ( N  \  { K }
) ) )
20 reseq1 5277 . . . . . . . . . . 11  |-  ( p  =  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  ->  ( p  |`  ( N  \  { K } ) )  =  ( ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  |`  ( N  \  { K } ) ) )
2120eqeq2d 2471 . . . . . . . . . 10  |-  ( p  =  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  ->  ( s  =  ( p  |`  ( N  \  { K }
) )  <->  s  =  ( ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  |`  ( N  \  { K } ) ) ) )
2221adantr 465 . . . . . . . . 9  |-  ( ( p  =  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  /\  ( ( N  e.  V  /\  K  e.  N )  /\  s  e.  S ) )  -> 
( s  =  ( p  |`  ( N  \  { K } ) )  <->  s  =  ( ( i  e.  N  |->  if ( i  =  K ,  K , 
( s `  i
) ) )  |`  ( N  \  { K } ) ) ) )
2319, 22mpbird 232 . . . . . . . 8  |-  ( ( p  =  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  /\  ( ( N  e.  V  /\  K  e.  N )  /\  s  e.  S ) )  -> 
s  =  ( p  |`  ( N  \  { K } ) ) )
2423ex 434 . . . . . . 7  |-  ( p  =  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  ->  ( ( ( N  e.  V  /\  K  e.  N )  /\  s  e.  S
)  ->  s  =  ( p  |`  ( N 
\  { K }
) ) ) )
2524adantl 466 . . . . . 6  |-  ( ( ( ( N  e.  V  /\  K  e.  N )  /\  s  e.  S )  /\  p  =  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) ) )  ->  ( (
( N  e.  V  /\  K  e.  N
)  /\  s  e.  S )  ->  s  =  ( p  |`  ( N  \  { K } ) ) ) )
2612, 25rspcimedv 3212 . . . . 5  |-  ( ( ( N  e.  V  /\  K  e.  N
)  /\  s  e.  S )  ->  (
( ( N  e.  V  /\  K  e.  N )  /\  s  e.  S )  ->  E. p  e.  Q  s  =  ( p  |`  ( N 
\  { K }
) ) ) )
2726pm2.43i 47 . . . 4  |-  ( ( ( N  e.  V  /\  K  e.  N
)  /\  s  e.  S )  ->  E. p  e.  Q  s  =  ( p  |`  ( N 
\  { K }
) ) )
284fvtresfn 5957 . . . . . . 7  |-  ( p  e.  Q  ->  ( H `  p )  =  ( p  |`  ( N  \  { K } ) ) )
2928eqeq2d 2471 . . . . . 6  |-  ( p  e.  Q  ->  (
s  =  ( H `
 p )  <->  s  =  ( p  |`  ( N 
\  { K }
) ) ) )
3029adantl 466 . . . . 5  |-  ( ( ( ( N  e.  V  /\  K  e.  N )  /\  s  e.  S )  /\  p  e.  Q )  ->  (
s  =  ( H `
 p )  <->  s  =  ( p  |`  ( N 
\  { K }
) ) ) )
3130rexbidva 2965 . . . 4  |-  ( ( ( N  e.  V  /\  K  e.  N
)  /\  s  e.  S )  ->  ( E. p  e.  Q  s  =  ( H `  p )  <->  E. p  e.  Q  s  =  ( p  |`  ( N 
\  { K }
) ) ) )
3227, 31mpbird 232 . . 3  |-  ( ( ( N  e.  V  /\  K  e.  N
)  /\  s  e.  S )  ->  E. p  e.  Q  s  =  ( H `  p ) )
3332ralrimiva 2871 . 2  |-  ( ( N  e.  V  /\  K  e.  N )  ->  A. s  e.  S  E. p  e.  Q  s  =  ( H `  p ) )
34 dffo3 6047 . 2  |-  ( H : Q -onto-> S  <->  ( H : Q --> S  /\  A. s  e.  S  E. p  e.  Q  s  =  ( H `  p ) ) )
356, 33, 34sylanbrc 664 1  |-  ( ( N  e.  V  /\  K  e.  N )  ->  H : Q -onto-> S
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811    \ cdif 3468   ifcif 3944   {csn 4032    |-> cmpt 4515    |` cres 5010   -->wf 5590   -onto->wfo 5592   ` 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-rep 4568  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:  symgfixf1o  16592
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