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Theorem symgfixfo 17092
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 17089 . . 3  |-  ( K  e.  N  ->  H : Q --> S )
65adantl 468 . 2  |-  ( ( N  e.  V  /\  K  e.  N )  ->  H : Q --> S )
7 eqeq1 2457 . . . . . . . . . 10  |-  ( i  =  j  ->  (
i  =  K  <->  j  =  K ) )
8 fveq2 5870 . . . . . . . . . 10  |-  ( i  =  j  ->  (
s `  i )  =  ( s `  j ) )
97, 8ifbieq2d 3908 . . . . . . . . 9  |-  ( i  =  j  ->  if ( i  =  K ,  K ,  ( s `  i ) )  =  if ( j  =  K ,  K ,  ( s `  j ) ) )
109cbvmptv 4498 . . . . . . . 8  |-  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  =  ( j  e.  N  |->  if ( j  =  K ,  K ,  ( s `  j ) ) )
111, 2, 3, 4, 10symgfixfolem1 17091 . . . . . . 7  |-  ( ( N  e.  V  /\  K  e.  N  /\  s  e.  S )  ->  ( i  e.  N  |->  if ( i  =  K ,  K , 
( s `  i
) ) )  e.  Q )
12113expa 1209 . . . . . 6  |-  ( ( ( N  e.  V  /\  K  e.  N
)  /\  s  e.  S )  ->  (
i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  e.  Q
)
13 simpr 463 . . . . . . . . . . . . 13  |-  ( ( N  e.  V  /\  K  e.  N )  ->  K  e.  N )
1413anim1i 572 . . . . . . . . . . . 12  |-  ( ( ( N  e.  V  /\  K  e.  N
)  /\  s  e.  S )  ->  ( K  e.  N  /\  s  e.  S )
)
1514adantl 468 . . . . . . . . . . 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 2453 . . . . . . . . . . . 12  |-  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  =  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )
173, 16symgextres 17078 . . . . . . . . . . 11  |-  ( ( K  e.  N  /\  s  e.  S )  ->  ( ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  |`  ( N  \  { K } ) )  =  s )
1815, 17syl 17 . . . . . . . . . 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 2459 . . . . . . . . 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 5102 . . . . . . . . . . 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 2463 . . . . . . . . . 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 467 . . . . . . . . 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 236 . . . . . . . 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 436 . . . . . . 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 468 . . . . . 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 3154 . . . . 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 49 . . . 4  |-  ( ( ( N  e.  V  /\  K  e.  N
)  /\  s  e.  S )  ->  E. p  e.  Q  s  =  ( p  |`  ( N 
\  { K }
) ) )
284fvtresfn 5955 . . . . . . 7  |-  ( p  e.  Q  ->  ( H `  p )  =  ( p  |`  ( N  \  { K } ) ) )
2928eqeq2d 2463 . . . . . 6  |-  ( p  e.  Q  ->  (
s  =  ( H `
 p )  <->  s  =  ( p  |`  ( N 
\  { K }
) ) ) )
3029adantl 468 . . . . 5  |-  ( ( ( ( N  e.  V  /\  K  e.  N )  /\  s  e.  S )  /\  p  e.  Q )  ->  (
s  =  ( H `
 p )  <->  s  =  ( p  |`  ( N 
\  { K }
) ) ) )
3130rexbidva 2900 . . . 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 236 . . 3  |-  ( ( ( N  e.  V  /\  K  e.  N
)  /\  s  e.  S )  ->  E. p  e.  Q  s  =  ( H `  p ) )
3332ralrimiva 2804 . 2  |-  ( ( N  e.  V  /\  K  e.  N )  ->  A. s  e.  S  E. p  e.  Q  s  =  ( H `  p ) )
34 dffo3 6042 . 2  |-  ( H : Q -onto-> S  <->  ( H : Q --> S  /\  A. s  e.  S  E. p  e.  Q  s  =  ( H `  p ) ) )
356, 33, 34sylanbrc 671 1  |-  ( ( N  e.  V  /\  K  e.  N )  ->  H : Q -onto-> S
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1446    e. wcel 1889   A.wral 2739   E.wrex 2740   {crab 2743    \ cdif 3403   ifcif 3883   {csn 3970    |-> cmpt 4464    |` cres 4839   -->wf 5581   -onto->wfo 5583   ` cfv 5585   Basecbs 15133   SymGrpcsymg 17030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-plusg 15215  df-tset 15221  df-symg 17031
This theorem is referenced by:  symgfixf1o  17093
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