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Theorem dihopelvalbN 31725
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 2408 . . . 4  |-  ( (
DIsoB `  K ) `  W )  =  ( ( DIsoB `  K ) `  W )
61, 2, 3, 4, 5dihvalb 31724 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( (
DIsoB `  K ) `  W ) `  X
) )
76eleq2d 2475 . 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 31635 . 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 245 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   <.cop 3781   class class class wbr 4176    e. cmpt 4230    _I cid 4457    |` cres 4843   ` cfv 5417   Basecbs 13428   lecple 13495   LHypclh 30470   LTrncltrn 30587   trLctrl 30644   DIsoBcdib 31625   DIsoHcdih 31715
This theorem is referenced by:  dihmeetlem1N  31777  dihglblem5apreN  31778  dihmeetlem4preN  31793
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-riota 6512  df-disoa 31516  df-dib 31626  df-dih 31716
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