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Theorem symgval 17069
Description: The value of the symmetric group function at  A. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.)
Hypotheses
Ref Expression
symgval.1  |-  G  =  ( SymGrp `  A )
symgval.2  |-  B  =  { x  |  x : A -1-1-onto-> A }
symgval.3  |-  .+  =  ( f  e.  B ,  g  e.  B  |->  ( f  o.  g
) )
symgval.4  |-  J  =  ( Xt_ `  ( A  X.  { ~P A } ) )
Assertion
Ref Expression
symgval  |-  ( A  e.  V  ->  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
Distinct variable group:    f, g, x, A
Allowed substitution hints:    B( x, f, g)    .+ ( x, f, g)    G( x, f, g)    J( x, f, g)    V( x, f, g)

Proof of Theorem symgval
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 symgval.1 . 2  |-  G  =  ( SymGrp `  A )
2 elex 3066 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
3 ovex 6343 . . . . . . 7  |-  ( a  ^m  a )  e. 
_V
4 f1of 5837 . . . . . . . . 9  |-  ( x : a -1-1-onto-> a  ->  x : a --> a )
5 vex 3060 . . . . . . . . . 10  |-  a  e. 
_V
65, 5elmap 7526 . . . . . . . . 9  |-  ( x  e.  ( a  ^m  a )  <->  x :
a --> a )
74, 6sylibr 217 . . . . . . . 8  |-  ( x : a -1-1-onto-> a  ->  x  e.  ( a  ^m  a
) )
87abssi 3516 . . . . . . 7  |-  { x  |  x : a -1-1-onto-> a } 
C_  ( a  ^m  a )
93, 8ssexi 4562 . . . . . 6  |-  { x  |  x : a -1-1-onto-> a }  e.  _V
109a1i 11 . . . . 5  |-  ( a  =  A  ->  { x  |  x : a -1-1-onto-> a }  e.  _V )
11 id 22 . . . . . . . 8  |-  ( b  =  { x  |  x : a -1-1-onto-> a }  ->  b  =  {
x  |  x : a -1-1-onto-> a } )
12 f1oeq23 5831 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  a  =  A )  ->  ( x : a -1-1-onto-> a  <-> 
x : A -1-1-onto-> A ) )
1312anidms 655 . . . . . . . . . 10  |-  ( a  =  A  ->  (
x : a -1-1-onto-> a  <->  x : A
-1-1-onto-> A ) )
1413abbidv 2580 . . . . . . . . 9  |-  ( a  =  A  ->  { x  |  x : a -1-1-onto-> a }  =  { x  |  x : A -1-1-onto-> A }
)
15 symgval.2 . . . . . . . . 9  |-  B  =  { x  |  x : A -1-1-onto-> A }
1614, 15syl6eqr 2514 . . . . . . . 8  |-  ( a  =  A  ->  { x  |  x : a -1-1-onto-> a }  =  B )
1711, 16sylan9eqr 2518 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  b  =  B )
1817opeq2d 4187 . . . . . 6  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  <. ( Base `  ndx ) ,  b
>.  =  <. ( Base `  ndx ) ,  B >. )
19 eqidd 2463 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( f  o.  g )  =  ( f  o.  g ) )
2017, 17, 19mpt2eq123dv 6380 . . . . . . . 8  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( f  e.  b ,  g  e.  b  |->  ( f  o.  g ) )  =  ( f  e.  B ,  g  e.  B  |->  ( f  o.  g
) ) )
21 symgval.3 . . . . . . . 8  |-  .+  =  ( f  e.  B ,  g  e.  B  |->  ( f  o.  g
) )
2220, 21syl6eqr 2514 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( f  e.  b ,  g  e.  b  |->  ( f  o.  g ) )  = 
.+  )
2322opeq2d 4187 . . . . . 6  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  <. ( +g  ` 
ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( f  o.  g ) ) >.  =  <. ( +g  `  ndx ) ,  .+  >. )
24 simpl 463 . . . . . . . . . 10  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  a  =  A )
2524pweqd 3968 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ~P a  =  ~P A )
2625sneqd 3992 . . . . . . . . . 10  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  { ~P a }  =  { ~P A } )
2724, 26xpeq12d 4878 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( a  X.  { ~P a } )  =  ( A  X.  { ~P A } ) )
2827fveq2d 5892 . . . . . . . 8  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( Xt_ `  ( a  X.  { ~P a } ) )  =  ( Xt_ `  ( A  X.  { ~P A } ) ) )
29 symgval.4 . . . . . . . 8  |-  J  =  ( Xt_ `  ( A  X.  { ~P A } ) )
3028, 29syl6eqr 2514 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( Xt_ `  ( a  X.  { ~P a } ) )  =  J )
3130opeq2d 4187 . . . . . 6  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  <. (TopSet `  ndx ) ,  ( Xt_ `  ( a  X.  { ~P a } ) )
>.  =  <. (TopSet `  ndx ) ,  J >. )
3218, 23, 31tpeq123d 4079 . . . . 5  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  ( f  e.  b ,  g  e.  b 
|->  ( f  o.  g
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( a  X.  { ~P a } ) ) >. }  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
3310, 32csbied 3402 . . . 4  |-  ( a  =  A  ->  [_ {
x  |  x : a -1-1-onto-> a }  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( f  o.  g ) ) >. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( a  X.  { ~P a } ) )
>. }  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (TopSet ` 
ndx ) ,  J >. } )
34 df-symg 17068 . . . 4  |-  SymGrp  =  ( a  e.  _V  |->  [_ { x  |  x : a -1-1-onto-> a }  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( f  o.  g ) ) >. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( a  X.  { ~P a } ) )
>. } )
35 tpex 6617 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (TopSet ` 
ndx ) ,  J >. }  e.  _V
3633, 34, 35fvmpt 5971 . . 3  |-  ( A  e.  _V  ->  ( SymGrp `
 A )  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
372, 36syl 17 . 2  |-  ( A  e.  V  ->  ( SymGrp `
 A )  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
381, 37syl5eq 2508 1  |-  ( A  e.  V  ->  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   {cab 2448   _Vcvv 3057   [_csb 3375   ~Pcpw 3963   {csn 3980   {ctp 3984   <.cop 3986    X. cxp 4851    o. ccom 4857   -->wf 5597   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6315    |-> cmpt2 6317    ^m cmap 7498   ndxcnx 15167   Basecbs 15170   +g cplusg 15239  TopSetcts 15245   Xt_cpt 15386   SymGrpcsymg 17067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-map 7500  df-symg 17068
This theorem is referenced by:  symgbas  17070  symgplusg  17079  symgtset  17089
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