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Theorem dibopelvalN 31941
Description: Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibval.b  |-  B  =  ( Base `  K
)
dibval.h  |-  H  =  ( LHyp `  K
)
dibval.t  |-  T  =  ( ( LTrn `  K
) `  W )
dibval.o  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
dibval.j  |-  J  =  ( ( DIsoA `  K
) `  W )
dibval.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibopelvalN  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  ( <. F ,  S >.  e.  ( I `  X
)  <->  ( F  e.  ( J `  X
)  /\  S  =  .0.  ) ) )
Distinct variable groups:    f, K    f, W    T, f
Allowed substitution hints:    B( f)    S( f)    F( f)    H( f)    I( f)    J( f)    V( f)    X( f)    .0. ( f)

Proof of Theorem dibopelvalN
StepHypRef Expression
1 dibval.b . . . 4  |-  B  =  ( Base `  K
)
2 dibval.h . . . 4  |-  H  =  ( LHyp `  K
)
3 dibval.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
4 dibval.o . . . 4  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
5 dibval.j . . . 4  |-  J  =  ( ( DIsoA `  K
) `  W )
6 dibval.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
71, 2, 3, 4, 5, 6dibval 31940 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
87eleq2d 2503 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  ( <. F ,  S >.  e.  ( I `  X
)  <->  <. F ,  S >.  e.  ( ( J `
 X )  X. 
{  .0.  } ) ) )
9 opelxp 4908 . . 3  |-  ( <. F ,  S >.  e.  ( ( J `  X )  X.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  e.  {  .0.  } ) )
10 fvex 5742 . . . . . . . 8  |-  ( (
LTrn `  K ) `  W )  e.  _V
113, 10eqeltri 2506 . . . . . . 7  |-  T  e. 
_V
1211mptex 5966 . . . . . 6  |-  ( f  e.  T  |->  (  _I  |`  B ) )  e. 
_V
134, 12eqeltri 2506 . . . . 5  |-  .0.  e.  _V
1413elsnc2 3843 . . . 4  |-  ( S  e.  {  .0.  }  <->  S  =  .0.  )
1514anbi2i 676 . . 3  |-  ( ( F  e.  ( J `
 X )  /\  S  e.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  =  .0.  ) )
169, 15bitri 241 . 2  |-  ( <. F ,  S >.  e.  ( ( J `  X )  X.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  =  .0.  ) )
178, 16syl6bb 253 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  ( <. F ,  S >.  e.  ( I `  X
)  <->  ( F  e.  ( J `  X
)  /\  S  =  .0.  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   {csn 3814   <.cop 3817    e. cmpt 4266    _I cid 4493    X. cxp 4876   dom cdm 4878    |` cres 4880   ` cfv 5454   Basecbs 13469   LHypclh 30781   LTrncltrn 30898   DIsoAcdia 31826   DIsoBcdib 31936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-dib 31937
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