Theorem dilfsetN 35439

Description: The mapping from fiducial atom to set of dilations. (Contributed by NM, 30-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dilset.a	|- A = ( Atoms ` K )
dilset.s	|- S = ( PSubSp ` K )
dilset.w	|- W = ( WAtoms ` K )
dilset.m	|- M = ( PAut ` K )
dilset.l	|- L = ( Dil ` K )

Assertion

Ref	Expression
dilfsetN	|- ( K e. B -> L = ( d e. A |-> { f e. M | A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) } ) )

Distinct variable groups:   A, d   f, d, x, K   f, M   x, S

Allowed substitution hints:   A( x, f)   B( x, f, d)   S( f, d)   L( x, f, d)   M( x, d)   W( x, f, d)

Proof of Theorem dilfsetN

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. B -> K e. _V )
2		dilset.l	. . 3 |- L = ( Dil ` K )
3		fveq2 6191	. . . . . 6 |- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) )
4		dilset.a	. . . . . 6 |- A = ( Atoms ` K )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( k = K -> ( Atoms ` k ) = A )
6		fveq2 6191	. . . . . . 7 |- ( k = K -> ( PAut ` k ) = ( PAut ` K ) )
7		dilset.m	. . . . . . 7 |- M = ( PAut ` K )
8	6, 7	syl6eqr 2674	. . . . . 6 |- ( k = K -> ( PAut ` k ) = M )
9		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( PSubSp ` k ) = ( PSubSp ` K ) )
10		dilset.s	. . . . . . . 8 |- S = ( PSubSp ` K )
11	9, 10	syl6eqr 2674	. . . . . . 7 |- ( k = K -> ( PSubSp ` k ) = S )
12		fveq2 6191	. . . . . . . . . . 11 |- ( k = K -> ( WAtoms ` k ) = ( WAtoms ` K ) )
13		dilset.w	. . . . . . . . . . 11 |- W = ( WAtoms ` K )
14	12, 13	syl6eqr 2674	. . . . . . . . . 10 |- ( k = K -> ( WAtoms ` k ) = W )
15	14	fveq1d 6193	. . . . . . . . 9 |- ( k = K -> ( ( WAtoms ` k ) ` d ) = ( W ` d ) )
16	15	sseq2d 3633	. . . . . . . 8 |- ( k = K -> ( x C_ ( ( WAtoms ` k ) ` d ) <-> x C_ ( W ` d ) ) )
17	16	imbi1d 331	. . . . . . 7 |- ( k = K -> ( ( x C_ ( ( WAtoms ` k ) ` d ) -> ( f ` x ) = x ) <-> ( x C_ ( W ` d ) -> ( f ` x ) = x ) ) )
18	11, 17	raleqbidv 3152	. . . . . 6 |- ( k = K -> ( A. x e. ( PSubSp ` k ) ( x C_ ( ( WAtoms ` k ) ` d ) -> ( f ` x ) = x ) <-> A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) ) )
19	8, 18	rabeqbidv 3195	. . . . 5 |- ( k = K -> { f e. ( PAut ` k ) | A. x e. ( PSubSp ` k ) ( x C_ ( ( WAtoms ` k ) ` d ) -> ( f ` x ) = x ) } = { f e. M | A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) } )
20	5, 19	mpteq12dv 4733	. . . 4 |- ( k = K -> ( d e. ( Atoms ` k ) |-> { f e. ( PAut ` k ) | A. x e. ( PSubSp ` k ) ( x C_ ( ( WAtoms ` k ) ` d ) -> ( f ` x ) = x ) } ) = ( d e. A |-> { f e. M | A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) } ) )
21		df-dilN 35392	. . . 4 |- Dil = ( k e. _V |-> ( d e. ( Atoms ` k ) |-> { f e. ( PAut ` k ) | A. x e. ( PSubSp ` k ) ( x C_ ( ( WAtoms ` k ) ` d ) -> ( f ` x ) = x ) } ) )
22		fvex 6201	. . . . . 6 |- ( Atoms ` K ) e. _V
23	4, 22	eqeltri 2697	. . . . 5 |- A e. _V
24	23	mptex 6486	. . . 4 |- ( d e. A |-> { f e. M | A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) } ) e. _V
25	20, 21, 24	fvmpt 6282	. . 3 |- ( K e. _V -> ( Dil ` K ) = ( d e. A |-> { f e. M | A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) } ) )
26	2, 25	syl5eq 2668	. 2 |- ( K e. _V -> L = ( d e. A |-> { f e. M | A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) } ) )
27	1, 26	syl 17	1 |- ( K e. B -> L = ( d e. A |-> { f e. M | A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   ` cfv 5888   Atomscatm 34550   PSubSpcpsubsp 34782   WAtomscwpointsN 35272   PAutcpautN 35273   DilcdilN 35388

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-res 5126  df-ima 5127  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-dilN 35392

This theorem is referenced by:  dilsetN  35440