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Theorem dibval2 31639
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 31531 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  J  <-> 
( X  e.  B  /\  X  .<_  W ) ) )
65biimpar 472 . 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 31637 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
116, 10syldan 457 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {csn 3782   class class class wbr 4180    e. cmpt 4234    _I cid 4461    X. cxp 4843   dom cdm 4845    |` cres 4847   ` cfv 5421   Basecbs 13432   lecple 13499   LHypclh 30478   LTrncltrn 30595   DIsoAcdia 31523   DIsoBcdib 31633
This theorem is referenced by:  dibopelval2  31640  dibval3N  31641  dibelval3  31642  dibelval1st  31644  dibelval2nd  31647  dibn0  31648  dibord  31654  dib0  31659  dib1dim  31660  dibss  31664  diblss  31665  dihwN  31784
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-13 1723  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-pow 4345  ax-pr 4371  ax-un 4668
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-pw 3769  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-disoa 31524  df-dib 31634
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