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Theorem dibfnN 36309
Description: Functionality and 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
dibfnN  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { x  e.  B  |  x  .<_  W } )
Distinct variable groups:    x,  .<_    x, B    x, K    x, W
Allowed substitution hints:    H( x)    I( x)    V( x)

Proof of Theorem dibfnN
StepHypRef Expression
1 dibfn.h . . 3  |-  H  =  ( LHyp `  K
)
2 eqid 2467 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
3 dibfn.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
41, 2, 3dibfna 36307 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  dom  (
( DIsoA `  K ) `  W ) )
5 dibfn.b . . . 4  |-  B  =  ( Base `  K
)
6 dibfn.l . . . 4  |-  .<_  =  ( le `  K )
75, 6, 1, 2diadm 36188 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  ( ( DIsoA `  K ) `  W
)  =  { x  e.  B  |  x  .<_  W } )
87fneq2d 5678 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( I  Fn  dom  ( ( DIsoA `  K
) `  W )  <->  I  Fn  { x  e.  B  |  x  .<_  W } ) )
94, 8mpbid 210 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { x  e.  B  |  x  .<_  W } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2821   class class class wbr 4453   dom cdm 5005    Fn wfn 5589   ` cfv 5594   Basecbs 14507   lecple 14579   LHypclh 35136   DIsoAcdia 36181   DIsoBcdib 36291
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-disoa 36182  df-dib 36292
This theorem is referenced by:  dibdmN  36310  dibf11N  36314
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