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Theorem dihopelvalbN 31480
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 2358 . . . 4  |-  ( (
DIsoB `  K ) `  W )  =  ( ( DIsoB `  K ) `  W )
61, 2, 3, 4, 5dihvalb 31479 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( (
DIsoB `  K ) `  W ) `  X
) )
76eleq2d 2425 . 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 31390 . 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 1642    e. wcel 1710   <.cop 3719   class class class wbr 4102    e. cmpt 4156    _I cid 4383    |` cres 4770   ` cfv 5334   Basecbs 13239   lecple 13306   LHypclh 30225   LTrncltrn 30342   trLctrl 30399   DIsoBcdib 31380   DIsoHcdih 31470
This theorem is referenced by:  dihmeetlem1N  31532  dihglblem5apreN  31533  dihmeetlem4preN  31548
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-riota 6388  df-disoa 31271  df-dib 31381  df-dih 31471
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