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Theorem dibffval 31330
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 2796 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5525 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dibval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2333 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5525 . . . . . . 7  |-  ( k  =  K  ->  ( DIsoA `  k )  =  ( DIsoA `  K )
)
65fveq1d 5527 . . . . . 6  |-  ( k  =  K  ->  (
( DIsoA `  k ) `  w )  =  ( ( DIsoA `  K ) `  w ) )
76dmeqd 4881 . . . . 5  |-  ( k  =  K  ->  dom  ( ( DIsoA `  k
) `  w )  =  dom  ( ( DIsoA `  K ) `  w
) )
86fveq1d 5527 . . . . . 6  |-  ( k  =  K  ->  (
( ( DIsoA `  k
) `  w ) `  x )  =  ( ( ( DIsoA `  K
) `  w ) `  x ) )
9 fveq2 5525 . . . . . . . . 9  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
109fveq1d 5527 . . . . . . . 8  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
11 fveq2 5525 . . . . . . . . . 10  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
12 dibval.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
1311, 12syl6eqr 2333 . . . . . . . . 9  |-  ( k  =  K  ->  ( Base `  k )  =  B )
1413reseq2d 4955 . . . . . . . 8  |-  ( k  =  K  ->  (  _I  |`  ( Base `  k
) )  =  (  _I  |`  B )
)
1510, 14mpteq12dv 4098 . . . . . . 7  |-  ( k  =  K  ->  (
f  e.  ( (
LTrn `  k ) `  w )  |->  (  _I  |`  ( Base `  k
) ) )  =  ( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) )
1615sneqd 3653 . . . . . 6  |-  ( k  =  K  ->  { ( f  e.  ( (
LTrn `  k ) `  w )  |->  (  _I  |`  ( Base `  k
) ) ) }  =  { ( f  e.  ( ( LTrn `  K ) `  w
)  |->  (  _I  |`  B ) ) } )
178, 16xpeq12d 4714 . . . . 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 4098 . . . 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 4098 . . 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 31329 . . 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 5539 . . . . 5  |-  ( LHyp `  K )  e.  _V
223, 21eqeltri 2353 . . . 4  |-  H  e. 
_V
2322mptex 5746 . . 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 5602 . 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 15 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 set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    e. cmpt 4077    _I cid 4304    X. cxp 4687   dom cdm 4689    |` cres 4691   ` cfv 5255   Basecbs 13148   LHypclh 30173   LTrncltrn 30290   DIsoAcdia 31218   DIsoBcdib 31328
This theorem is referenced by:  dibfval  31331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-dib 31329
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