Theorem symgval 16958

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 3026	. . 3 |- ( A e. V -> A e. _V )
3		ovex 6272	. . . . . . 7 |- ( a ^m a ) e. _V
4		f1of 5769	. . . . . . . . 9 |- ( x : a -1-1-onto-> a -> x : a --> a )
5		vex 3020	. . . . . . . . . 10 |- a e. _V
6	5, 5	elmap 7450	. . . . . . . . 9 |- ( x e. ( a ^m a ) <-> x : a --> a )
7	4, 6	sylibr 215	. . . . . . . 8 |- ( x : a -1-1-onto-> a -> x e. ( a ^m a ) )
8	7	abssi 3474	. . . . . . 7 |- { x | x : a -1-1-onto-> a } C_ ( a ^m a )
9	3, 8	ssexi 4507	. . . . . 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 5763	. . . . . . . . . . 11 |- ( ( a = A /\ a = A ) -> ( x : a -1-1-onto-> a <-> x : A -1-1-onto-> A ) )
13	12	anidms 649	. . . . . . . . . 10 |- ( a = A -> ( x : a -1-1-onto-> a <-> x : A -1-1-onto-> A ) )
14	13	abbidv 2541	. . . . . . . . 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 2475	. . . . . . . 8 |- ( a = A -> { x | x : a -1-1-onto-> a } = B )
17	11, 16	sylan9eqr 2479	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> b = B )
18	17	opeq2d 4132	. . . . . 6 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> <. ( Base ` ndx ) , b >. = <. ( Base ` ndx ) , B >. )
19		eqidd 2424	. . . . . . . . 9 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( f o. g ) = ( f o. g ) )
20	17, 17, 19	mpt2eq123dv 6306	. . . . . . . 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 2475	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( f e. b , g e. b |-> ( f o. g ) ) = .+ )
23	22	opeq2d 4132	. . . . . 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 458	. . . . . . . . . 10 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> a = A )
25	24	pweqd 3924	. . . . . . . . . . 11 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ~P a = ~P A )
26	25	sneqd 3948	. . . . . . . . . 10 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> { ~P a } = { ~P A } )
27	24, 26	xpeq12d 4816	. . . . . . . . 9 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( a X. { ~P a } ) = ( A X. { ~P A } ) )
28	27	fveq2d 5824	. . . . . . . 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 2475	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( Xt_ ` ( a X. { ~P a } ) ) = J )
31	30	opeq2d 4132	. . . . . 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 4032	. . . . 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 3360	. . . 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 16957	. . . 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 6543	. . . 4 |- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. } e. _V
36	33, 34, 35	fvmpt 5903	. . 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 2469	1 |- ( A e. V -> G = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. } )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1872   {cab 2409   _Vcvv 3017   [_csb 3333   ~Pcpw 3919   {csn 3936   {ctp 3940   <.cop 3942   X. cxp 4789   o. ccom 4795   -->wf 5535   -1-1-onto->wf1o 5538   ` cfv 5539  (class class class)co 6244   |-> cmpt2 6246   ^m cmap 7422   ndxcnx 15056   Basecbs 15059   +g cplusg 15128  TopSetcts 15134   Xt_cpt 15275   SymGrpcsymg 16956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-br 4362  df-opab 4421  df-mpt 4422  df-id 4706  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-map 7424  df-symg 16957

This theorem is referenced by:  symgbas  16959  symgplusg  16968  symgtset  16978