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Theorem dibffval 31399
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 2872 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5608 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dibval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2408 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5608 . . . . . . 7  |-  ( k  =  K  ->  ( DIsoA `  k )  =  ( DIsoA `  K )
)
65fveq1d 5610 . . . . . 6  |-  ( k  =  K  ->  (
( DIsoA `  k ) `  w )  =  ( ( DIsoA `  K ) `  w ) )
76dmeqd 4963 . . . . 5  |-  ( k  =  K  ->  dom  ( ( DIsoA `  k
) `  w )  =  dom  ( ( DIsoA `  K ) `  w
) )
86fveq1d 5610 . . . . . 6  |-  ( k  =  K  ->  (
( ( DIsoA `  k
) `  w ) `  x )  =  ( ( ( DIsoA `  K
) `  w ) `  x ) )
9 fveq2 5608 . . . . . . . . 9  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
109fveq1d 5610 . . . . . . . 8  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
11 fveq2 5608 . . . . . . . . . 10  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
12 dibval.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
1311, 12syl6eqr 2408 . . . . . . . . 9  |-  ( k  =  K  ->  ( Base `  k )  =  B )
1413reseq2d 5037 . . . . . . . 8  |-  ( k  =  K  ->  (  _I  |`  ( Base `  k
) )  =  (  _I  |`  B )
)
1510, 14mpteq12dv 4179 . . . . . . 7  |-  ( k  =  K  ->  (
f  e.  ( (
LTrn `  k ) `  w )  |->  (  _I  |`  ( Base `  k
) ) )  =  ( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) )
1615sneqd 3729 . . . . . 6  |-  ( k  =  K  ->  { ( f  e.  ( (
LTrn `  k ) `  w )  |->  (  _I  |`  ( Base `  k
) ) ) }  =  { ( f  e.  ( ( LTrn `  K ) `  w
)  |->  (  _I  |`  B ) ) } )
178, 16xpeq12d 4796 . . . . 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 4179 . . . 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 4179 . . 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 31398 . . 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 5622 . . . . 5  |-  ( LHyp `  K )  e.  _V
223, 21eqeltri 2428 . . . 4  |-  H  e. 
_V
2322mptex 5832 . . 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 5685 . 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 1642    e. wcel 1710   _Vcvv 2864   {csn 3716    e. cmpt 4158    _I cid 4386    X. cxp 4769   dom cdm 4771    |` cres 4773   ` cfv 5337   Basecbs 13245   LHypclh 30242   LTrncltrn 30359   DIsoAcdia 31287   DIsoBcdib 31397
This theorem is referenced by:  dibfval  31400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-dib 31398
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