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Theorem dihopelvalbN 31428
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 2283 . . . 4  |-  ( (
DIsoB `  K ) `  W )  =  ( ( DIsoB `  K ) `  W )
61, 2, 3, 4, 5dihvalb 31427 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( (
DIsoB `  K ) `  W ) `  X
) )
76eleq2d 2350 . 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 31338 . 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 244 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023    e. cmpt 4077    _I cid 4304    |` cres 4691   ` cfv 5255   Basecbs 13148   lecple 13215   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   DIsoBcdib 31328   DIsoHcdih 31418
This theorem is referenced by:  dihmeetlem1N  31480  dihglblem5apreN  31481  dihmeetlem4preN  31496
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-ov 5861  df-riota 6304  df-disoa 31219  df-dib 31329  df-dih 31419
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