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Theorem dihopelvalbN 32134
Description: Ordered pair member of the partial isomorphism H for argument under  W. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihval3.b  |-  B  =  ( Base `  K
)
dihval3.l  |-  .<_  =  ( le `  K )
dihval3.h  |-  H  =  ( LHyp `  K
)
dihval3.t  |-  T  =  ( ( LTrn `  K
) `  W )
dihval3.r  |-  R  =  ( ( trL `  K
) `  W )
dihval3.o  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
dihval3.i  |-  I  =  ( ( DIsoH `  K
) `  W )
Assertion
Ref Expression
dihopelvalbN  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( <. F ,  S >.  e.  ( I `  X
)  <->  ( ( F  e.  T  /\  ( R `  F )  .<_  X )  /\  S  =  O ) ) )
Distinct variable groups:    g, K    T, g    g, W
Allowed substitution hints:    B( g)    R( g)    S( g)    F( g)    H( g)    I( g)    .<_ ( g)    O( g)    V( g)    X( g)

Proof of Theorem dihopelvalbN
StepHypRef Expression
1 dihval3.b . . . 4  |-  B  =  ( Base `  K
)
2 dihval3.l . . . 4  |-  .<_  =  ( le `  K )
3 dihval3.h . . . 4  |-  H  =  ( LHyp `  K
)
4 dihval3.i . . . 4  |-  I  =  ( ( DIsoH `  K
) `  W )
5 eqid 2442 . . . 4  |-  ( (
DIsoB `  K ) `  W )  =  ( ( DIsoB `  K ) `  W )
61, 2, 3, 4, 5dihvalb 32133 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( (
DIsoB `  K ) `  W ) `  X
) )
76eleq2d 2509 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( <. F ,  S >.  e.  ( I `  X
)  <->  <. F ,  S >.  e.  ( ( (
DIsoB `  K ) `  W ) `  X
) ) )
8 dihval3.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
9 dihval3.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
10 dihval3.o . . 3  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
111, 2, 3, 8, 9, 10, 5dibopelval3 32044 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( <. F ,  S >.  e.  ( ( ( DIsoB `  K ) `  W
) `  X )  <->  ( ( F  e.  T  /\  ( R `  F
)  .<_  X )  /\  S  =  O )
) )
127, 11bitrd 246 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( <. F ,  S >.  e.  ( I `  X
)  <->  ( ( F  e.  T  /\  ( R `  F )  .<_  X )  /\  S  =  O ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727   <.cop 3841   class class class wbr 4237    e. cmpt 4291    _I cid 4522    |` cres 4909   ` cfv 5483   Basecbs 13500   lecple 13567   LHypclh 30879   LTrncltrn 30996   trLctrl 31053   DIsoBcdib 32034   DIsoHcdih 32124
This theorem is referenced by:  dihmeetlem1N  32186  dihglblem5apreN  32187  dihmeetlem4preN  32202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-riota 6578  df-disoa 31925  df-dib 32035  df-dih 32125
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