Theorem symgval 15877

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 2979	. . 3 |- ( A e. V -> A e. _V )
3		ovex 6115	. . . . . . 7 |- ( a ^m a ) e. _V
4		f1of 5638	. . . . . . . . 9 |- ( x : a -1-1-onto-> a -> x : a --> a )
5		vex 2973	. . . . . . . . . 10 |- a e. _V
6	5, 5	elmap 7237	. . . . . . . . 9 |- ( x e. ( a ^m a ) <-> x : a --> a )
7	4, 6	sylibr 212	. . . . . . . 8 |- ( x : a -1-1-onto-> a -> x e. ( a ^m a ) )
8	7	abssi 3424	. . . . . . 7 |- { x | x : a -1-1-onto-> a } C_ ( a ^m a )
9	3, 8	ssexi 4434	. . . . . 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 5632	. . . . . . . . . . 11 |- ( ( a = A /\ a = A ) -> ( x : a -1-1-onto-> a <-> x : A -1-1-onto-> A ) )
13	12	anidms 640	. . . . . . . . . 10 |- ( a = A -> ( x : a -1-1-onto-> a <-> x : A -1-1-onto-> A ) )
14	13	abbidv 2555	. . . . . . . . 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 2491	. . . . . . . 8 |- ( a = A -> { x | x : a -1-1-onto-> a } = B )
17	11, 16	sylan9eqr 2495	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> b = B )
18	17	opeq2d 4063	. . . . . 6 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> <. ( Base ` ndx ) , b >. = <. ( Base ` ndx ) , B >. )
19		eqidd 2442	. . . . . . . . 9 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( f o. g ) = ( f o. g ) )
20	17, 17, 19	mpt2eq123dv 6147	. . . . . . . 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 2491	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( f e. b , g e. b |-> ( f o. g ) ) = .+ )
23	22	opeq2d 4063	. . . . . 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 454	. . . . . . . . . 10 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> a = A )
25	24	pweqd 3862	. . . . . . . . . . 11 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ~P a = ~P A )
26	25	sneqd 3886	. . . . . . . . . 10 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> { ~P a } = { ~P A } )
27	24, 26	xpeq12d 4861	. . . . . . . . 9 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( a X. { ~P a } ) = ( A X. { ~P A } ) )
28	27	fveq2d 5692	. . . . . . . 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 2491	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( Xt_ ` ( a X. { ~P a } ) ) = J )
31	30	opeq2d 4063	. . . . . 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 3966	. . . . 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 3311	. . . 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 15876	. . . 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 6378	. . . 4 |- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. } e. _V
36	33, 34, 35	fvmpt 5771	. . 3 |- ( A e. _V -> ( SymGrp ` A ) = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. } )
37	2, 36	syl 16	. 2 |- ( A e. V -> ( SymGrp ` A ) = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. } )
38	1, 37	syl5eq 2485	1 |- ( A e. V -> G = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. } )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1364   e. wcel 1761   {cab 2427   _Vcvv 2970   [_csb 3285   ~Pcpw 3857   {csn 3874   {ctp 3878   <.cop 3880   X. cxp 4834   o. ccom 4840   -->wf 5411   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   e. cmpt2 6092   ^m cmap 7210   ndxcnx 14167   Basecbs 14170   +g cplusg 14234  TopSetcts 14240   Xt_cpt 14373   SymGrpcsymg 15875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7212  df-symg 15876

This theorem is referenced by:  symgbas  15878  symgplusg  15887  symgtset  15897