Theorem symgval 17799

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 3212	. . 3 |- ( A e. V -> A e. _V )
3		ovex 6678	. . . . . . 7 |- ( a ^m a ) e. _V
4		f1of 6137	. . . . . . . . 9 |- ( x : a -1-1-onto-> a -> x : a --> a )
5		vex 3203	. . . . . . . . . 10 |- a e. _V
6	5, 5	elmap 7886	. . . . . . . . 9 |- ( x e. ( a ^m a ) <-> x : a --> a )
7	4, 6	sylibr 224	. . . . . . . 8 |- ( x : a -1-1-onto-> a -> x e. ( a ^m a ) )
8	7	abssi 3677	. . . . . . 7 |- { x | x : a -1-1-onto-> a } C_ ( a ^m a )
9	3, 8	ssexi 4803	. . . . . 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 6130	. . . . . . . . . . 11 |- ( ( a = A /\ a = A ) -> ( x : a -1-1-onto-> a <-> x : A -1-1-onto-> A ) )
13	12	anidms 677	. . . . . . . . . 10 |- ( a = A -> ( x : a -1-1-onto-> a <-> x : A -1-1-onto-> A ) )
14	13	abbidv 2741	. . . . . . . . 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 2674	. . . . . . . 8 |- ( a = A -> { x | x : a -1-1-onto-> a } = B )
17	11, 16	sylan9eqr 2678	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> b = B )
18	17	opeq2d 4409	. . . . . 6 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> <. ( Base ` ndx ) , b >. = <. ( Base ` ndx ) , B >. )
19		eqidd 2623	. . . . . . . . 9 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( f o. g ) = ( f o. g ) )
20	17, 17, 19	mpt2eq123dv 6717	. . . . . . . 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 2674	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( f e. b , g e. b |-> ( f o. g ) ) = .+ )
23	22	opeq2d 4409	. . . . . 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 473	. . . . . . . . . 10 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> a = A )
25	24	pweqd 4163	. . . . . . . . . . 11 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ~P a = ~P A )
26	25	sneqd 4189	. . . . . . . . . 10 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> { ~P a } = { ~P A } )
27	24, 26	xpeq12d 5140	. . . . . . . . 9 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( a X. { ~P a } ) = ( A X. { ~P A } ) )
28	27	fveq2d 6195	. . . . . . . 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 2674	. . . . . . 7 |- ( ( a = A /\ b = { x | x : a -1-1-onto-> a } ) -> ( Xt_ ` ( a X. { ~P a } ) ) = J )
31	30	opeq2d 4409	. . . . . 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 4283	. . . . 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 3560	. . . 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 17798	. . . 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 6957	. . . 4 |- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. } e. _V
36	33, 34, 35	fvmpt 6282	. . 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 2668	1 |- ( A e. V -> G = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   [_csb 3533   ~Pcpw 4158   {csn 4177   {ctp 4181   <.cop 4183   X. cxp 5112   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   ndxcnx 15854   Basecbs 15857   +g cplusg 15941  TopSetcts 15947   Xt_cpt 16099   SymGrpcsymg 17797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-symg 17798

This theorem is referenced by:  symgbas  17800  symgplusg  17809  symgtset  17819