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Theorem symgval 14787
Description: The value of the symmetry 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 2809 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
3 ovex 5899 . . . . . . 7  |-  ( a  ^m  a )  e. 
_V
4 f1of 5488 . . . . . . . . 9  |-  ( x : a -1-1-onto-> a  ->  x : a --> a )
5 vex 2804 . . . . . . . . . 10  |-  a  e. 
_V
65, 5elmap 6812 . . . . . . . . 9  |-  ( x  e.  ( a  ^m  a )  <->  x :
a --> a )
74, 6sylibr 203 . . . . . . . 8  |-  ( x : a -1-1-onto-> a  ->  x  e.  ( a  ^m  a
) )
87abssi 3261 . . . . . . 7  |-  { x  |  x : a -1-1-onto-> a } 
C_  ( a  ^m  a )
93, 8ssexi 4175 . . . . . 6  |-  { x  |  x : a -1-1-onto-> a }  e.  _V
109a1i 10 . . . . 5  |-  ( a  =  A  ->  { x  |  x : a -1-1-onto-> a }  e.  _V )
11 id 19 . . . . . . . 8  |-  ( b  =  { x  |  x : a -1-1-onto-> a }  ->  b  =  {
x  |  x : a -1-1-onto-> a } )
12 f1oeq23 5482 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  a  =  A )  ->  ( x : a -1-1-onto-> a  <-> 
x : A -1-1-onto-> A ) )
1312anidms 626 . . . . . . . . . 10  |-  ( a  =  A  ->  (
x : a -1-1-onto-> a  <->  x : A
-1-1-onto-> A ) )
1413abbidv 2410 . . . . . . . . 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 2346 . . . . . . . 8  |-  ( a  =  A  ->  { x  |  x : a -1-1-onto-> a }  =  B )
1711, 16sylan9eqr 2350 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  b  =  B )
1817opeq2d 3819 . . . . . 6  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  <. ( Base `  ndx ) ,  b
>.  =  <. ( Base `  ndx ) ,  B >. )
19 eqidd 2297 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( f  o.  g )  =  ( f  o.  g ) )
2017, 17, 19mpt2eq123dv 5926 . . . . . . . 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 2346 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( f  e.  b ,  g  e.  b  |->  ( f  o.  g ) )  = 
.+  )
2322opeq2d 3819 . . . . . 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 443 . . . . . . . . . 10  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  a  =  A )
2524pweqd 3643 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ~P a  =  ~P A )
2625sneqd 3666 . . . . . . . . . 10  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  { ~P a }  =  { ~P A } )
2724, 26xpeq12d 4730 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( a  X.  { ~P a } )  =  ( A  X.  { ~P A } ) )
2827fveq2d 5545 . . . . . . . 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 2346 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  ( Xt_ `  ( a  X.  { ~P a } ) )  =  J )
3130opeq2d 3819 . . . . . 6  |-  ( ( a  =  A  /\  b  =  { x  |  x : a -1-1-onto-> a } )  ->  <. (TopSet `  ndx ) ,  ( Xt_ `  ( a  X.  { ~P a } ) )
>.  =  <. (TopSet `  ndx ) ,  J >. )
3218, 23, 31tpeq123d 3734 . . . . 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 3136 . . . 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 14786 . . . 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 4535 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (TopSet ` 
ndx ) ,  J >. }  e.  _V
3633, 34, 35fvmpt 5618 . . 3  |-  ( A  e.  _V  ->  ( SymGrp `
 A )  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
372, 36syl 15 . 2  |-  ( A  e.  V  ->  ( SymGrp `
 A )  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
381, 37syl5eq 2340 1  |-  ( A  e.  V  ->  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   [_csb 3094   ~Pcpw 3638   {csn 3653   {ctp 3655   <.cop 3656    X. cxp 4703    o. ccom 4709   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    ^m cmap 6788   ndxcnx 13161   Basecbs 13164   +g cplusg 13224  TopSetcts 13230   Xt_cpt 13359   SymGrpcsymg 14785
This theorem is referenced by:  symgbas  14788  symgplusg  14792  symgtset  14795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-symg 14786
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