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.

Step	Hyp	Ref	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
6	5, 5	elmap 7526	. . . . . . . . 9 |- ( x e. ( a ^m a ) <-> x : a --> a )
7	4, 6	sylibr 217	. . . . . . . 8 |- ( x : a -1-1-onto-> a -> x e. ( a ^m a ) )
8	7	abssi 3516	. . . . . . 7 |- { x | x : a -1-1-onto-> a } C_ ( a ^m a )
9	3, 8	ssexi 4562	. . . . . 6 |- { x | x : a -1-1-onto-> a } e. _V
10	9	a1i 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 ) )
13	12	anidms 655	. . . . . . . . . 10 |- ( a = A -> ( x : a -1-1-onto-> a <-> x : A -1-1-onto-> A ) )
14	13	abbidv 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 }
16	14, 15	syl6eqr 2514	. . . . . . . 8 |- ( a = A -> { x | x : a -1-1-onto-> a } = B )
17	11, 16	sylan9eqr 2518	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> b = B )
18	17	opeq2d 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 ) )
20	17, 17, 19	mpt2eq123dv 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 ) )
22	20, 21	syl6eqr 2514	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( f e. b , g e. b |-> ( f o. g ) ) = .+ )
23	22	opeq2d 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 )
25	24	pweqd 3968	. . . . . . . . . . 11 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ~P a = ~P A )
26	25	sneqd 3992	. . . . . . . . . 10 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> { ~P a } = { ~P A } )
27	24, 26	xpeq12d 4878	. . . . . . . . 9 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( a X. { ~P a } ) = ( A X. { ~P A } ) )
28	27	fveq2d 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 } ) )
30	28, 29	syl6eqr 2514	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( Xt_ ` ( a X. { ~P a } ) ) = J )
31	30	opeq2d 4187	. . . . . 6 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> <. (TopSet ` ndx ) , ( Xt_ ` ( a X. { ~P a } ) ) >. = <. (TopSet ` ndx ) , J >. )
32	18, 23, 31	tpeq123d 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 >. } )
33	10, 32	csbied 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
36	33, 34, 35	fvmpt 5971	. . 3 |- ( A e. _V -> ( SymGrp ` A ) = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. } )
37	2, 36	syl 17	. 2 |- ( A e. V -> ( SymGrp ` A ) = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. } )
38	1, 37	syl5eq 2508	1 |- ( A e. V -> G = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. } )

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