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Theorem dibval2 34786
Description: Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.)
Hypotheses
Ref Expression
dibval2.b  |-  B  =  ( Base `  K
)
dibval2.l  |-  .<_  =  ( le `  K )
dibval2.h  |-  H  =  ( LHyp `  K
)
dibval2.t  |-  T  =  ( ( LTrn `  K
) `  W )
dibval2.o  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
dibval2.j  |-  J  =  ( ( DIsoA `  K
) `  W )
dibval2.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibval2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
Distinct variable groups:    f, K    f, W
Allowed substitution hints:    B( f)    T( f)    H( f)    I( f)    J( f)    .<_ ( f)    V( f)    X( f)    .0. ( f)

Proof of Theorem dibval2
StepHypRef Expression
1 dibval2.b . . . 4  |-  B  =  ( Base `  K
)
2 dibval2.l . . . 4  |-  .<_  =  ( le `  K )
3 dibval2.h . . . 4  |-  H  =  ( LHyp `  K
)
4 dibval2.j . . . 4  |-  J  =  ( ( DIsoA `  K
) `  W )
51, 2, 3, 4diaeldm 34678 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  J  <-> 
( X  e.  B  /\  X  .<_  W ) ) )
65biimpar 485 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  X  e.  dom  J )
7 dibval2.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
8 dibval2.o . . 3  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
9 dibval2.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
101, 3, 7, 8, 4, 9dibval 34784 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
116, 10syldan 470 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3875   class class class wbr 4290    e. cmpt 4348    _I cid 4629    X. cxp 4836   dom cdm 4838    |` cres 4840   ` cfv 5416   Basecbs 14172   lecple 14243   LHypclh 33625   LTrncltrn 33742   DIsoAcdia 34670   DIsoBcdib 34780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-disoa 34671  df-dib 34781
This theorem is referenced by:  dibopelval2  34787  dibval3N  34788  dibelval3  34789  dibelval1st  34791  dibelval2nd  34794  dibn0  34795  dibord  34801  dib0  34806  dib1dim  34807  dibss  34811  diblss  34812  dihwN  34931
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