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Theorem dibeldmN 31407
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 2366 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
3 dibfn.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
41, 2, 3dibdiadm 31404 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  dom  ( ( DIsoA `  K
) `  W )
)
54eleq2d 2433 . 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 31285 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  ( ( DIsoA `  K
) `  W )  <->  ( X  e.  B  /\  X  .<_  W ) ) )
95, 8bitrd 244 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  ( X  e.  B  /\  X  .<_  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   class class class wbr 4125   dom cdm 4792   ` cfv 5358   Basecbs 13356   lecple 13423   LHypclh 30232   DIsoAcdia 31277   DIsoBcdib 31387
This theorem is referenced by:  dibf11N  31410  dibintclN  31416  dihmeetlem2N  31548
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-disoa 31278  df-dib 31388
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