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Theorem dibffval 36338
Description: The partial isomorphism B for a lattice  K. (Contributed by NM, 8-Dec-2013.)
Hypotheses
Ref Expression
dibval.b  |-  B  =  ( Base `  K
)
dibval.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
dibffval  |-  ( K  e.  V  ->  ( DIsoB `  K )  =  ( w  e.  H  |->  ( x  e.  dom  ( ( DIsoA `  K
) `  w )  |->  ( ( ( (
DIsoA `  K ) `  w ) `  x
)  X.  { ( f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) ) } ) ) ) )
Distinct variable groups:    w, H    w, f, x, K
Allowed substitution hints:    B( x, w, f)    H( x, f)    V( x, w, f)

Proof of Theorem dibffval
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 3127 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5872 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dibval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2526 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5872 . . . . . . 7  |-  ( k  =  K  ->  ( DIsoA `  k )  =  ( DIsoA `  K )
)
65fveq1d 5874 . . . . . 6  |-  ( k  =  K  ->  (
( DIsoA `  k ) `  w )  =  ( ( DIsoA `  K ) `  w ) )
76dmeqd 5211 . . . . 5  |-  ( k  =  K  ->  dom  ( ( DIsoA `  k
) `  w )  =  dom  ( ( DIsoA `  K ) `  w
) )
86fveq1d 5874 . . . . . 6  |-  ( k  =  K  ->  (
( ( DIsoA `  k
) `  w ) `  x )  =  ( ( ( DIsoA `  K
) `  w ) `  x ) )
9 fveq2 5872 . . . . . . . . 9  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
109fveq1d 5874 . . . . . . . 8  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
11 fveq2 5872 . . . . . . . . . 10  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
12 dibval.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
1311, 12syl6eqr 2526 . . . . . . . . 9  |-  ( k  =  K  ->  ( Base `  k )  =  B )
1413reseq2d 5279 . . . . . . . 8  |-  ( k  =  K  ->  (  _I  |`  ( Base `  k
) )  =  (  _I  |`  B )
)
1510, 14mpteq12dv 4531 . . . . . . 7  |-  ( k  =  K  ->  (
f  e.  ( (
LTrn `  k ) `  w )  |->  (  _I  |`  ( Base `  k
) ) )  =  ( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) )
1615sneqd 4045 . . . . . 6  |-  ( k  =  K  ->  { ( f  e.  ( (
LTrn `  k ) `  w )  |->  (  _I  |`  ( Base `  k
) ) ) }  =  { ( f  e.  ( ( LTrn `  K ) `  w
)  |->  (  _I  |`  B ) ) } )
178, 16xpeq12d 5030 . . . . 5  |-  ( k  =  K  ->  (
( ( ( DIsoA `  k ) `  w
) `  x )  X.  { ( f  e.  ( ( LTrn `  k
) `  w )  |->  (  _I  |`  ( Base `  k ) ) ) } )  =  ( ( ( (
DIsoA `  K ) `  w ) `  x
)  X.  { ( f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) ) } ) )
187, 17mpteq12dv 4531 . . . 4  |-  ( k  =  K  ->  (
x  e.  dom  (
( DIsoA `  k ) `  w )  |->  ( ( ( ( DIsoA `  k
) `  w ) `  x )  X.  {
( f  e.  ( ( LTrn `  k
) `  w )  |->  (  _I  |`  ( Base `  k ) ) ) } ) )  =  ( x  e. 
dom  ( ( DIsoA `  K ) `  w
)  |->  ( ( ( ( DIsoA `  K ) `  w ) `  x
)  X.  { ( f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) ) } ) ) )
194, 18mpteq12dv 4531 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  ( x  e.  dom  ( (
DIsoA `  k ) `  w )  |->  ( ( ( ( DIsoA `  k
) `  w ) `  x )  X.  {
( f  e.  ( ( LTrn `  k
) `  w )  |->  (  _I  |`  ( Base `  k ) ) ) } ) ) )  =  ( w  e.  H  |->  ( x  e.  dom  ( (
DIsoA `  K ) `  w )  |->  ( ( ( ( DIsoA `  K
) `  w ) `  x )  X.  {
( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) } ) ) ) )
20 df-dib 36337 . . 3  |-  DIsoB  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  dom  ( (
DIsoA `  k ) `  w )  |->  ( ( ( ( DIsoA `  k
) `  w ) `  x )  X.  {
( f  e.  ( ( LTrn `  k
) `  w )  |->  (  _I  |`  ( Base `  k ) ) ) } ) ) ) )
21 fvex 5882 . . . . 5  |-  ( LHyp `  K )  e.  _V
223, 21eqeltri 2551 . . . 4  |-  H  e. 
_V
2322mptex 6142 . . 3  |-  ( w  e.  H  |->  ( x  e.  dom  ( (
DIsoA `  K ) `  w )  |->  ( ( ( ( DIsoA `  K
) `  w ) `  x )  X.  {
( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) } ) ) )  e.  _V
2419, 20, 23fvmpt 5957 . 2  |-  ( K  e.  _V  ->  ( DIsoB `  K )  =  ( w  e.  H  |->  ( x  e.  dom  ( ( DIsoA `  K
) `  w )  |->  ( ( ( (
DIsoA `  K ) `  w ) `  x
)  X.  { ( f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) ) } ) ) ) )
251, 24syl 16 1  |-  ( K  e.  V  ->  ( DIsoB `  K )  =  ( w  e.  H  |->  ( x  e.  dom  ( ( DIsoA `  K
) `  w )  |->  ( ( ( (
DIsoA `  K ) `  w ) `  x
)  X.  { ( f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) ) } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3118   {csn 4033    |-> cmpt 4511    _I cid 4796    X. cxp 5003   dom cdm 5005    |` cres 5007   ` cfv 5594   Basecbs 14507   LHypclh 35181   LTrncltrn 35298   DIsoAcdia 36226   DIsoBcdib 36336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-dib 36337
This theorem is referenced by:  dibfval  36339
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