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Theorem symgtset 16564
Description: The topology of the symmetric group on  A. This component is defined on a larger set than the true base - the product topology is defined on the set of all functions, not just bijections - but the definition of  TopOpen ensures that it is trimmed down before it gets use. (Contributed by Mario Carneiro, 29-Aug-2015.)
Hypothesis
Ref Expression
symggrp.1  |-  G  =  ( SymGrp `  A )
Assertion
Ref Expression
symgtset  |-  ( A  e.  V  ->  ( Xt_ `  ( A  X.  { ~P A } ) )  =  (TopSet `  G ) )

Proof of Theorem symgtset
Dummy variables  f 
g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 symggrp.1 . . . 4  |-  G  =  ( SymGrp `  A )
2 eqid 2396 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
31, 2symgbas 16545 . . . 4  |-  ( Base `  G )  =  {
x  |  x : A -1-1-onto-> A }
4 eqid 2396 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
51, 2, 4symgplusg 16554 . . . 4  |-  ( +g  `  G )  =  ( f  e.  ( Base `  G ) ,  g  e.  ( Base `  G
)  |->  ( f  o.  g ) )
6 eqid 2396 . . . 4  |-  ( Xt_ `  ( A  X.  { ~P A } ) )  =  ( Xt_ `  ( A  X.  { ~P A } ) )
71, 3, 5, 6symgval 16544 . . 3  |-  ( A  e.  V  ->  G  =  { <. ( Base `  ndx ) ,  ( Base `  G ) >. ,  <. ( +g  `  ndx ) ,  ( +g  `  G
) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( A  X.  { ~P A }
) ) >. } )
87fveq2d 5795 . 2  |-  ( A  e.  V  ->  (TopSet `  G )  =  (TopSet `  { <. ( Base `  ndx ) ,  ( Base `  G ) >. ,  <. ( +g  `  ndx ) ,  ( +g  `  G
) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( A  X.  { ~P A }
) ) >. } ) )
9 fvex 5801 . . 3  |-  ( Xt_ `  ( A  X.  { ~P A } ) )  e.  _V
10 eqid 2396 . . . 4  |-  { <. (
Base `  ndx ) ,  ( Base `  G
) >. ,  <. ( +g  `  ndx ) ,  ( +g  `  G
) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( A  X.  { ~P A }
) ) >. }  =  { <. ( Base `  ndx ) ,  ( Base `  G ) >. ,  <. ( +g  `  ndx ) ,  ( +g  `  G
) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( A  X.  { ~P A }
) ) >. }
1110topgrptset 14821 . . 3  |-  ( (
Xt_ `  ( A  X.  { ~P A }
) )  e.  _V  ->  ( Xt_ `  ( A  X.  { ~P A } ) )  =  (TopSet `  { <. ( Base `  ndx ) ,  ( Base `  G
) >. ,  <. ( +g  `  ndx ) ,  ( +g  `  G
) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( A  X.  { ~P A }
) ) >. } ) )
129, 11ax-mp 5 . 2  |-  ( Xt_ `  ( A  X.  { ~P A } ) )  =  (TopSet `  { <. ( Base `  ndx ) ,  ( Base `  G ) >. ,  <. ( +g  `  ndx ) ,  ( +g  `  G
) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( A  X.  { ~P A }
) ) >. } )
138, 12syl6reqr 2456 1  |-  ( A  e.  V  ->  ( Xt_ `  ( A  X.  { ~P A } ) )  =  (TopSet `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1836   _Vcvv 3051   ~Pcpw 3944   {csn 3961   {ctp 3965   <.cop 3967    X. cxp 4928   ` cfv 5513   ndxcnx 14654   Basecbs 14657   +g cplusg 14725  TopSetcts 14731   Xt_cpt 14869   SymGrpcsymg 16542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-1st 6721  df-2nd 6722  df-recs 6982  df-rdg 7016  df-1o 7070  df-oadd 7074  df-er 7251  df-map 7362  df-en 7458  df-dom 7459  df-sdom 7460  df-fin 7461  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-nn 10475  df-2 10533  df-3 10534  df-4 10535  df-5 10536  df-6 10537  df-7 10538  df-8 10539  df-9 10540  df-n0 10735  df-z 10804  df-uz 11024  df-fz 11616  df-struct 14659  df-ndx 14660  df-slot 14661  df-base 14662  df-plusg 14738  df-tset 14744  df-symg 16543
This theorem is referenced by:  symgtopn  16570
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