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Theorem dihopelvalbN 37378
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 2382 . . . 4  |-  ( (
DIsoB `  K ) `  W )  =  ( ( DIsoB `  K ) `  W )
61, 2, 3, 4, 5dihvalb 37377 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( (
DIsoB `  K ) `  W ) `  X
) )
76eleq2d 2452 . 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 37288 . 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 253 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   <.cop 3950   class class class wbr 4367    |-> cmpt 4425    _I cid 4704    |` cres 4915   ` cfv 5496   Basecbs 14634   lecple 14709   LHypclh 36121   LTrncltrn 36238   trLctrl 36296   DIsoBcdib 37278   DIsoHcdih 37368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-disoa 37169  df-dib 37279  df-dih 37369
This theorem is referenced by:  dihmeetlem1N  37430  dihglblem5apreN  37431  dihmeetlem4preN  37446
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