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Theorem symgval 16958
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 3026 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
3 ovex 6272 . . . . . . 7  |-  ( a  ^m  a )  e. 
_V
4 f1of 5769 . . . . . . . . 9  |-  ( x : a -1-1-onto-> a  ->  x : a --> a )
5 vex 3020 . . . . . . . . . 10  |-  a  e. 
_V
65, 5elmap 7450 . . . . . . . . 9  |-  ( x  e.  ( a  ^m  a )  <->  x :
a --> a )
74, 6sylibr 215 . . . . . . . 8  |-  ( x : a -1-1-onto-> a  ->  x  e.  ( a  ^m  a
) )
87abssi 3474 . . . . . . 7  |-  { x  |  x : a -1-1-onto-> a } 
C_  ( a  ^m  a )
93, 8ssexi 4507 . . . . . 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 5763 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  a  =  A )  ->  ( x : a -1-1-onto-> a  <-> 
x : A -1-1-onto-> A ) )
1312anidms 649 . . . . . . . . . 10  |-  ( a  =  A  ->  (
x : a -1-1-onto-> a  <->  x : A
-1-1-onto-> A ) )
1413abbidv 2541 . . . . . . . . 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 2475 . . . . . . . 8  |-  ( a  =  A  ->  { x  |  x : a -1-1-onto-> a }  =  B )
1711, 16sylan9eqr 2479 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  b  =  B )
1817opeq2d 4132 . . . . . 6  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  <. ( Base `  ndx ) ,  b
>.  =  <. ( Base `  ndx ) ,  B >. )
19 eqidd 2424 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( f  o.  g )  =  ( f  o.  g ) )
2017, 17, 19mpt2eq123dv 6306 . . . . . . . 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 2475 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( f  e.  b ,  g  e.  b  |->  ( f  o.  g ) )  = 
.+  )
2322opeq2d 4132 . . . . . 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 458 . . . . . . . . . 10  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  a  =  A )
2524pweqd 3924 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ~P a  =  ~P A )
2625sneqd 3948 . . . . . . . . . 10  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  { ~P a }  =  { ~P A } )
2724, 26xpeq12d 4816 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( a  X.  { ~P a } )  =  ( A  X.  { ~P A } ) )
2827fveq2d 5824 . . . . . . . 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 2475 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( Xt_ `  ( a  X.  { ~P a } ) )  =  J )
3130opeq2d 4132 . . . . . 6  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  <. (TopSet `  ndx ) ,  ( Xt_ `  ( a  X.  { ~P a } ) )
>.  =  <. (TopSet `  ndx ) ,  J >. )
3218, 23, 31tpeq123d 4032 . . . . 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 3360 . . . 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 16957 . . . 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 6543 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (TopSet ` 
ndx ) ,  J >. }  e.  _V
3633, 34, 35fvmpt 5903 . . 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 2469 1  |-  ( A  e.  V  ->  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   {cab 2409   _Vcvv 3017   [_csb 3333   ~Pcpw 3919   {csn 3936   {ctp 3940   <.cop 3942    X. cxp 4789    o. ccom 4795   -->wf 5535   -1-1-onto->wf1o 5538   ` cfv 5539  (class class class)co 6244    |-> cmpt2 6246    ^m cmap 7422   ndxcnx 15056   Basecbs 15059   +g cplusg 15128  TopSetcts 15134   Xt_cpt 15275   SymGrpcsymg 16956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-br 4362  df-opab 4421  df-mpt 4422  df-id 4706  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-map 7424  df-symg 16957
This theorem is referenced by:  symgbas  16959  symgplusg  16968  symgtset  16978
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