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Theorem dibeldmN 35955
Description: Member of domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibfn.b  |-  B  =  ( Base `  K
)
dibfn.l  |-  .<_  =  ( le `  K )
dibfn.h  |-  H  =  ( LHyp `  K
)
dibfn.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibeldmN  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  ( X  e.  B  /\  X  .<_  W ) ) )

Proof of Theorem dibeldmN
StepHypRef Expression
1 dibfn.h . . . 4  |-  H  =  ( LHyp `  K
)
2 eqid 2467 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
3 dibfn.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
41, 2, 3dibdiadm 35952 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  dom  ( ( DIsoA `  K
) `  W )
)
54eleq2d 2537 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  X  e.  dom  (
( DIsoA `  K ) `  W ) ) )
6 dibfn.b . . 3  |-  B  =  ( Base `  K
)
7 dibfn.l . . 3  |-  .<_  =  ( le `  K )
86, 7, 1, 2diaeldm 35833 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  ( ( DIsoA `  K
) `  W )  <->  ( X  e.  B  /\  X  .<_  W ) ) )
95, 8bitrd 253 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  ( X  e.  B  /\  X  .<_  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4447   dom cdm 4999   ` cfv 5586   Basecbs 14483   lecple 14555   LHypclh 34780   DIsoAcdia 35825   DIsoBcdib 35935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-disoa 35826  df-dib 35936
This theorem is referenced by:  dibf11N  35958  dibintclN  35964  dihmeetlem2N  36096
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