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Theorem dibffval 31635
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 2932 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5695 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dibval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2462 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5695 . . . . . . 7  |-  ( k  =  K  ->  ( DIsoA `  k )  =  ( DIsoA `  K )
)
65fveq1d 5697 . . . . . 6  |-  ( k  =  K  ->  (
( DIsoA `  k ) `  w )  =  ( ( DIsoA `  K ) `  w ) )
76dmeqd 5039 . . . . 5  |-  ( k  =  K  ->  dom  ( ( DIsoA `  k
) `  w )  =  dom  ( ( DIsoA `  K ) `  w
) )
86fveq1d 5697 . . . . . 6  |-  ( k  =  K  ->  (
( ( DIsoA `  k
) `  w ) `  x )  =  ( ( ( DIsoA `  K
) `  w ) `  x ) )
9 fveq2 5695 . . . . . . . . 9  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
109fveq1d 5697 . . . . . . . 8  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
11 fveq2 5695 . . . . . . . . . 10  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
12 dibval.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
1311, 12syl6eqr 2462 . . . . . . . . 9  |-  ( k  =  K  ->  ( Base `  k )  =  B )
1413reseq2d 5113 . . . . . . . 8  |-  ( k  =  K  ->  (  _I  |`  ( Base `  k
) )  =  (  _I  |`  B )
)
1510, 14mpteq12dv 4255 . . . . . . 7  |-  ( k  =  K  ->  (
f  e.  ( (
LTrn `  k ) `  w )  |->  (  _I  |`  ( Base `  k
) ) )  =  ( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) )
1615sneqd 3795 . . . . . 6  |-  ( k  =  K  ->  { ( f  e.  ( (
LTrn `  k ) `  w )  |->  (  _I  |`  ( Base `  k
) ) ) }  =  { ( f  e.  ( ( LTrn `  K ) `  w
)  |->  (  _I  |`  B ) ) } )
178, 16xpeq12d 4870 . . . . 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 4255 . . . 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 4255 . . 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 31634 . . 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 5709 . . . . 5  |-  ( LHyp `  K )  e.  _V
223, 21eqeltri 2482 . . . 4  |-  H  e. 
_V
2322mptex 5933 . . 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 5773 . 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 1649    e. wcel 1721   _Vcvv 2924   {csn 3782    e. cmpt 4234    _I cid 4461    X. cxp 4843   dom cdm 4845    |` cres 4847   ` cfv 5421   Basecbs 13432   LHypclh 30478   LTrncltrn 30595   DIsoAcdia 31523   DIsoBcdib 31633
This theorem is referenced by:  dibfval  31636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-dib 31634
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