Theorem symgval 15889

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 2986	. . 3 |- ( A e. V -> A e. _V )
3		ovex 6121	. . . . . . 7 |- ( a ^m a ) e. _V
4		f1of 5646	. . . . . . . . 9 |- ( x : a -1-1-onto-> a -> x : a --> a )
5		vex 2980	. . . . . . . . . 10 |- a e. _V
6	5, 5	elmap 7246	. . . . . . . . 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 3432	. . . . . . 7 |- { x | x : a -1-1-onto-> a } C_ ( a ^m a )
9	3, 8	ssexi 4442	. . . . . 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 5640	. . . . . . . . . . 11 |- ( ( a = A /\ a = A ) -> ( x : a -1-1-onto-> a <-> x : A -1-1-onto-> A ) )
13	12	anidms 645	. . . . . . . . . 10 |- ( a = A -> ( x : a -1-1-onto-> a <-> x : A -1-1-onto-> A ) )
14	13	abbidv 2562	. . . . . . . . 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 2493	. . . . . . . 8 |- ( a = A -> { x | x : a -1-1-onto-> a } = B )
17	11, 16	sylan9eqr 2497	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> b = B )
18	17	opeq2d 4071	. . . . . 6 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> <. ( Base ` ndx ) , b >. = <. ( Base ` ndx ) , B >. )
19		eqidd 2444	. . . . . . . . 9 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( f o. g ) = ( f o. g ) )
20	17, 17, 19	mpt2eq123dv 6153	. . . . . . . 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 2493	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( f e. b , g e. b |-> ( f o. g ) ) = .+ )
23	22	opeq2d 4071	. . . . . 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 457	. . . . . . . . . 10 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> a = A )
25	24	pweqd 3870	. . . . . . . . . . 11 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ~P a = ~P A )
26	25	sneqd 3894	. . . . . . . . . 10 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> { ~P a } = { ~P A } )
27	24, 26	xpeq12d 4870	. . . . . . . . 9 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( a X. { ~P a } ) = ( A X. { ~P A } ) )
28	27	fveq2d 5700	. . . . . . . 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 2493	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( Xt_ ` ( a X. { ~P a } ) ) = J )
31	30	opeq2d 4071	. . . . . 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 3974	. . . . 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 3319	. . . 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 15888	. . . 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 6384	. . . 4 |- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. } e. _V
36	33, 34, 35	fvmpt 5779	. . 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 2487	1 |- ( A e. V -> G = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. } )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   {cab 2429   _Vcvv 2977   [_csb 3293   ~Pcpw 3865   {csn 3882   {ctp 3886   <.cop 3888   X. cxp 4843   o. ccom 4849   -->wf 5419   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   e. cmpt2 6098   ^m cmap 7219   ndxcnx 14176   Basecbs 14179   +g cplusg 14243  TopSetcts 14249   Xt_cpt 14382   SymGrpcsymg 15887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-map 7221  df-symg 15888

This theorem is referenced by:  symgbas  15890  symgplusg  15899  symgtset  15909