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Theorem dibopelvalN 30706
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 30705 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
87eleq2d 2350 . 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 4719 . . 3  |-  ( <. F ,  S >.  e.  ( ( J `  X )  X.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  e.  {  .0.  } ) )
10 fvex 5539 . . . . . . . 8  |-  ( (
LTrn `  K ) `  W )  e.  _V
113, 10eqeltri 2353 . . . . . . 7  |-  T  e. 
_V
1211mptex 5746 . . . . . 6  |-  ( f  e.  T  |->  (  _I  |`  B ) )  e. 
_V
134, 12eqeltri 2353 . . . . 5  |-  .0.  e.  _V
1413elsnc2 3669 . . . 4  |-  ( S  e.  {  .0.  }  <->  S  =  .0.  )
1514anbi2i 675 . . 3  |-  ( ( F  e.  ( J `
 X )  /\  S  e.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  =  .0.  ) )
169, 15bitri 240 . 2  |-  ( <. F ,  S >.  e.  ( ( J `  X )  X.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  =  .0.  ) )
178, 16syl6bb 252 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   <.cop 3643    e. cmpt 4077    _I cid 4304    X. cxp 4687   dom cdm 4689    |` cres 4691   ` cfv 5255   Basecbs 13148   LHypclh 29546   LTrncltrn 29663   DIsoAcdia 30591   DIsoBcdib 30701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-dib 30702
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