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Theorem ldilfset 33744
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 3040 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 fveq2 5879 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 ldilset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2523 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5879 . . . . . 6  |-  ( k  =  K  ->  ( LAut `  k )  =  ( LAut `  K
) )
6 ldilset.i . . . . . 6  |-  I  =  ( LAut `  K
)
75, 6syl6eqr 2523 . . . . 5  |-  ( k  =  K  ->  ( LAut `  k )  =  I )
8 fveq2 5879 . . . . . . 7  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
9 ldilset.b . . . . . . 7  |-  B  =  ( Base `  K
)
108, 9syl6eqr 2523 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  B )
11 fveq2 5879 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
12 ldilset.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
1311, 12syl6eqr 2523 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1413breqd 4406 . . . . . . 7  |-  ( k  =  K  ->  (
x ( le `  k ) w  <->  x  .<_  w ) )
1514imbi1d 324 . . . . . 6  |-  ( k  =  K  ->  (
( x ( le
`  k ) w  ->  ( f `  x )  =  x )  <->  ( x  .<_  w  ->  ( f `  x )  =  x ) ) )
1610, 15raleqbidv 2987 . . . . 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 3026 . . . 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 4474 . . 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 33740 . . 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 5889 . . . . 5  |-  ( LHyp `  K )  e.  _V
213, 20eqeltri 2545 . . . 4  |-  H  e. 
_V
2221mptex 6152 . . 3  |-  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  (
x  .<_  w  ->  (
f `  x )  =  x ) } )  e.  _V
2318, 19, 22fvmpt 5963 . 2  |-  ( K  e.  _V  ->  ( LDil `  K )  =  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) } ) )
241, 23syl 17 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 1452    e. wcel 1904   A.wral 2756   {crab 2760   _Vcvv 3031   class class class wbr 4395    |-> cmpt 4454   ` cfv 5589   Basecbs 15199   lecple 15275   LHypclh 33620   LAutclaut 33621   LDilcldil 33736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ldil 33740
This theorem is referenced by:  ldilset  33745
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