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Theorem symgfixfo 15944
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 15941 . . 3  |-  ( K  e.  N  ->  H : Q --> S )
65adantl 466 . 2  |-  ( ( N  e.  V  /\  K  e.  N )  ->  H : Q --> S )
7 eqeq1 2448 . . . . . . . . . 10  |-  ( i  =  j  ->  (
i  =  K  <->  j  =  K ) )
8 fveq2 5690 . . . . . . . . . 10  |-  ( i  =  j  ->  (
s `  i )  =  ( s `  j ) )
97, 8ifbieq2d 3813 . . . . . . . . 9  |-  ( i  =  j  ->  if ( i  =  K ,  K ,  ( s `  i ) )  =  if ( j  =  K ,  K ,  ( s `  j ) ) )
109cbvmptv 4382 . . . . . . . 8  |-  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  =  ( j  e.  N  |->  if ( j  =  K ,  K ,  ( s `  j ) ) )
111, 2, 3, 4, 10symgfixfolem1 15943 . . . . . . 7  |-  ( ( N  e.  V  /\  K  e.  N  /\  s  e.  S )  ->  ( i  e.  N  |->  if ( i  =  K ,  K , 
( s `  i
) ) )  e.  Q )
12113expa 1187 . . . . . 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 2442 . . . . . . . . . . . 12  |-  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  =  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )
173, 16symgextres 15929 . . . . . . . . . . 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 2447 . . . . . . . . 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 5103 . . . . . . . . . . 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 2453 . . . . . . . . . 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 3074 . . . . 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 5774 . . . . . . 7  |-  ( p  e.  Q  ->  ( H `  p )  =  ( p  |`  ( N  \  { K } ) ) )
2928eqeq2d 2453 . . . . . 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 2731 . . . 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 2798 . 2  |-  ( ( N  e.  V  /\  K  e.  N )  ->  A. s  e.  S  E. p  e.  Q  s  =  ( H `  p ) )
34 dffo3 5857 . 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 1369    e. wcel 1756   A.wral 2714   E.wrex 2715   {crab 2718    \ cdif 3324   ifcif 3790   {csn 3876    e. cmpt 4349    |` cres 4841   -->wf 5413   -onto->wfo 5415   ` cfv 5417   Basecbs 14173   SymGrpcsymg 15881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-plusg 14250  df-tset 14256  df-symg 15882
This theorem is referenced by:  symgfixf1o  15945
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