Theorem ldilfset 35394

Description: The mapping from fiducial co-atom w to its set of lattice dilations. (Contributed by NM, 11-May-2012.)

Hypotheses

Ref	Expression
ldilset.b	|- B = ( Base ` K )
ldilset.l	|- .<_ = ( le ` K )
ldilset.h	|- H = ( LHyp ` K )
ldilset.i	|- I = ( LAut ` K )

Assertion

Ref	Expression
ldilfset	|- ( K e. C -> ( LDil ` K ) = ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } ) )

Distinct variable groups:   x, B   w, H   f, I   w, f, x, K

Allowed substitution hints:   B( w, f)   C( x, w, f)   H( x, f)   I( x, w)   .<_ ( x, w, f)

Proof of Theorem ldilfset

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. C -> K e. _V )
2		fveq2 6191	. . . . 5 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
3		ldilset.h	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2674	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 6191	. . . . . 6 |- ( k = K -> ( LAut ` k ) = ( LAut ` K ) )
6		ldilset.i	. . . . . 6 |- I = ( LAut ` K )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( k = K -> ( LAut ` k ) = I )
8		fveq2 6191	. . . . . . 7 |- ( k = K -> ( Base ` k ) = ( Base ` K ) )
9		ldilset.b	. . . . . . 7 |- B = ( Base ` K )
10	8, 9	syl6eqr 2674	. . . . . 6 |- ( k = K -> ( Base ` k ) = B )
11		fveq2 6191	. . . . . . . . 9 |- ( k = K -> ( le ` k ) = ( le ` K ) )
12		ldilset.l	. . . . . . . . 9 |- .<_ = ( le ` K )
13	11, 12	syl6eqr 2674	. . . . . . . 8 |- ( k = K -> ( le ` k ) = .<_ )
14	13	breqd 4664	. . . . . . 7 |- ( k = K -> ( x ( le ` k ) w <-> x .<_ w ) )
15	14	imbi1d 331	. . . . . 6 |- ( k = K -> ( ( x ( le ` k ) w -> ( f ` x ) = x ) <-> ( x .<_ w -> ( f ` x ) = x ) ) )
16	10, 15	raleqbidv 3152	. . . . 5 |- ( k = K -> ( A. x e. ( Base ` k ) ( x ( le ` k ) w -> ( f ` x ) = x ) <-> A. x e. B ( x .<_ w -> ( f ` x ) = x ) ) )
17	7, 16	rabeqbidv 3195	. . . 4 |- ( k = K -> { f e. ( LAut ` k ) | A. x e. ( Base ` k ) ( x ( le ` k ) w -> ( f ` x ) = x ) } = { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } )
18	4, 17	mpteq12dv 4733	. . 3 |- ( k = K -> ( w e. ( LHyp ` k ) |-> { f e. ( LAut ` k ) | A. x e. ( Base ` k ) ( x ( le ` k ) w -> ( f ` x ) = x ) } ) = ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } ) )
19		df-ldil 35390	. . 3 |- LDil = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { f e. ( LAut ` k ) | A. x e. ( Base ` k ) ( x ( le ` k ) w -> ( f ` x ) = x ) } ) )
20		fvex 6201	. . . . 5 |- ( LHyp ` K ) e. _V
21	3, 20	eqeltri 2697	. . . 4 |- H e. _V
22	21	mptex 6486	. . 3 |- ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } ) e. _V
23	18, 19, 22	fvmpt 6282	. 2 |- ( K e. _V -> ( LDil ` K ) = ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } ) )
24	1, 23	syl 17	1 |- ( K e. C -> ( LDil ` K ) = ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888   Basecbs 15857   lecple 15948   LHypclh 35270   LAutclaut 35271   LDilcldil 35386

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-ldil 35390

This theorem is referenced by:  ldilset  35395