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Theorem ldilfset 30905
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.
StepHypRef Expression
1 elex 2964 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 fveq2 5728 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 ldilset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2486 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5728 . . . . . 6  |-  ( k  =  K  ->  ( LAut `  k )  =  ( LAut `  K
) )
6 ldilset.i . . . . . 6  |-  I  =  ( LAut `  K
)
75, 6syl6eqr 2486 . . . . 5  |-  ( k  =  K  ->  ( LAut `  k )  =  I )
8 fveq2 5728 . . . . . . 7  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
9 ldilset.b . . . . . . 7  |-  B  =  ( Base `  K
)
108, 9syl6eqr 2486 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  B )
11 fveq2 5728 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
12 ldilset.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
1311, 12syl6eqr 2486 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1413breqd 4223 . . . . . . 7  |-  ( k  =  K  ->  (
x ( le `  k ) w  <->  x  .<_  w ) )
1514imbi1d 309 . . . . . 6  |-  ( k  =  K  ->  (
( x ( le
`  k ) w  ->  ( f `  x )  =  x )  <->  ( x  .<_  w  ->  ( f `  x )  =  x ) ) )
1610, 15raleqbidv 2916 . . . . 5  |-  ( k  =  K  ->  ( A. x  e.  ( Base `  k ) ( x ( le `  k ) w  -> 
( f `  x
)  =  x )  <->  A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) ) )
177, 16rabeqbidv 2951 . . . 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 ) } )
184, 17mpteq12dv 4287 . . 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 30901 . . 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 5742 . . . . 5  |-  ( LHyp `  K )  e.  _V
213, 20eqeltri 2506 . . . 4  |-  H  e. 
_V
2221mptex 5966 . . 3  |-  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  (
x  .<_  w  ->  (
f `  x )  =  x ) } )  e.  _V
2318, 19, 22fvmpt 5806 . 2  |-  ( K  e.  _V  ->  ( LDil `  K )  =  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) } ) )
241, 23syl 16 1  |-  ( K  e.  C  ->  ( LDil `  K )  =  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956   class class class wbr 4212    e. cmpt 4266   ` cfv 5454   Basecbs 13469   lecple 13536   LHypclh 30781   LAutclaut 30782   LDilcldil 30897
This theorem is referenced by:  ldilset  30906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ldil 30901
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