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Theorem symgfixfo 16255
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 16252 . . 3  |-  ( K  e.  N  ->  H : Q --> S )
65adantl 466 . 2  |-  ( ( N  e.  V  /\  K  e.  N )  ->  H : Q --> S )
7 eqeq1 2466 . . . . . . . . . 10  |-  ( i  =  j  ->  (
i  =  K  <->  j  =  K ) )
8 fveq2 5859 . . . . . . . . . 10  |-  ( i  =  j  ->  (
s `  i )  =  ( s `  j ) )
97, 8ifbieq2d 3959 . . . . . . . . 9  |-  ( i  =  j  ->  if ( i  =  K ,  K ,  ( s `  i ) )  =  if ( j  =  K ,  K ,  ( s `  j ) ) )
109cbvmptv 4533 . . . . . . . 8  |-  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  =  ( j  e.  N  |->  if ( j  =  K ,  K ,  ( s `  j ) ) )
111, 2, 3, 4, 10symgfixfolem1 16254 . . . . . . 7  |-  ( ( N  e.  V  /\  K  e.  N  /\  s  e.  S )  ->  ( i  e.  N  |->  if ( i  =  K ,  K , 
( s `  i
) ) )  e.  Q )
12113expa 1191 . . . . . 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 2462 . . . . . . . . . . . 12  |-  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )  =  ( i  e.  N  |->  if ( i  =  K ,  K ,  ( s `  i ) ) )
173, 16symgextres 16240 . . . . . . . . . . 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 2470 . . . . . . . . 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 5260 . . . . . . . . . . 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 2476 . . . . . . . . . 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 3211 . . . . 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 5944 . . . . . . 7  |-  ( p  e.  Q  ->  ( H `  p )  =  ( p  |`  ( N  \  { K } ) ) )
2928eqeq2d 2476 . . . . . 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 2873 . 2  |-  ( ( N  e.  V  /\  K  e.  N )  ->  A. s  e.  S  E. p  e.  Q  s  =  ( H `  p ) )
34 dffo3 6029 . 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 1374    e. wcel 1762   A.wral 2809   E.wrex 2810   {crab 2813    \ cdif 3468   ifcif 3934   {csn 4022    |-> cmpt 4500    |` cres 4996   -->wf 5577   -onto->wfo 5579   ` cfv 5581   Basecbs 14481   SymGrpcsymg 16192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-plusg 14559  df-tset 14565  df-symg 16193
This theorem is referenced by:  symgfixf1o  16256
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