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Theorem dibffval 32012
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 2966 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5731 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dibval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2488 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5731 . . . . . . 7  |-  ( k  =  K  ->  ( DIsoA `  k )  =  ( DIsoA `  K )
)
65fveq1d 5733 . . . . . 6  |-  ( k  =  K  ->  (
( DIsoA `  k ) `  w )  =  ( ( DIsoA `  K ) `  w ) )
76dmeqd 5075 . . . . 5  |-  ( k  =  K  ->  dom  ( ( DIsoA `  k
) `  w )  =  dom  ( ( DIsoA `  K ) `  w
) )
86fveq1d 5733 . . . . . 6  |-  ( k  =  K  ->  (
( ( DIsoA `  k
) `  w ) `  x )  =  ( ( ( DIsoA `  K
) `  w ) `  x ) )
9 fveq2 5731 . . . . . . . . 9  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
109fveq1d 5733 . . . . . . . 8  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
11 fveq2 5731 . . . . . . . . . 10  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
12 dibval.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
1311, 12syl6eqr 2488 . . . . . . . . 9  |-  ( k  =  K  ->  ( Base `  k )  =  B )
1413reseq2d 5149 . . . . . . . 8  |-  ( k  =  K  ->  (  _I  |`  ( Base `  k
) )  =  (  _I  |`  B )
)
1510, 14mpteq12dv 4290 . . . . . . 7  |-  ( k  =  K  ->  (
f  e.  ( (
LTrn `  k ) `  w )  |->  (  _I  |`  ( Base `  k
) ) )  =  ( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) )
1615sneqd 3829 . . . . . 6  |-  ( k  =  K  ->  { ( f  e.  ( (
LTrn `  k ) `  w )  |->  (  _I  |`  ( Base `  k
) ) ) }  =  { ( f  e.  ( ( LTrn `  K ) `  w
)  |->  (  _I  |`  B ) ) } )
178, 16xpeq12d 4906 . . . . 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 4290 . . . 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 4290 . . 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 32011 . . 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 5745 . . . . 5  |-  ( LHyp `  K )  e.  _V
223, 21eqeltri 2508 . . . 4  |-  H  e. 
_V
2322mptex 5969 . . 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 5809 . 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 set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816    e. cmpt 4269    _I cid 4496    X. cxp 4879   dom cdm 4881    |` cres 4883   ` cfv 5457   Basecbs 13474   LHypclh 30855   LTrncltrn 30972   DIsoAcdia 31900   DIsoBcdib 32010
This theorem is referenced by:  dibfval  32013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-dib 32011
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