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Theorem ldilfset 33752
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 2981 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 fveq2 5691 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 ldilset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2493 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5691 . . . . . 6  |-  ( k  =  K  ->  ( LAut `  k )  =  ( LAut `  K
) )
6 ldilset.i . . . . . 6  |-  I  =  ( LAut `  K
)
75, 6syl6eqr 2493 . . . . 5  |-  ( k  =  K  ->  ( LAut `  k )  =  I )
8 fveq2 5691 . . . . . . 7  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
9 ldilset.b . . . . . . 7  |-  B  =  ( Base `  K
)
108, 9syl6eqr 2493 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  B )
11 fveq2 5691 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
12 ldilset.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
1311, 12syl6eqr 2493 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1413breqd 4303 . . . . . . 7  |-  ( k  =  K  ->  (
x ( le `  k ) w  <->  x  .<_  w ) )
1514imbi1d 317 . . . . . 6  |-  ( k  =  K  ->  (
( x ( le
`  k ) w  ->  ( f `  x )  =  x )  <->  ( x  .<_  w  ->  ( f `  x )  =  x ) ) )
1610, 15raleqbidv 2931 . . . . 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 2967 . . . 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 4370 . . 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 33748 . . 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 5701 . . . . 5  |-  ( LHyp `  K )  e.  _V
213, 20eqeltri 2513 . . . 4  |-  H  e. 
_V
2221mptex 5948 . . 3  |-  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  (
x  .<_  w  ->  (
f `  x )  =  x ) } )  e.  _V
2318, 19, 22fvmpt 5774 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2715   {crab 2719   _Vcvv 2972   class class class wbr 4292    e. cmpt 4350   ` cfv 5418   Basecbs 14174   lecple 14245   LHypclh 33628   LAutclaut 33629   LDilcldil 33744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ldil 33748
This theorem is referenced by:  ldilset  33753
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