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Theorem dibclN 35130
Description: Closure of partial isomorphism B for a lattice  K. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibcl.h  |-  H  =  ( LHyp `  K
)
dibcl.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibclN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
I `  X )  e.  ran  I )

Proof of Theorem dibclN
StepHypRef Expression
1 dibcl.h . . . 4  |-  H  =  ( LHyp `  K
)
2 eqid 2454 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
3 dibcl.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
41, 2, 3dibfna 35122 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  dom  (
( DIsoA `  K ) `  W ) )
5 fnfun 5615 . . 3  |-  ( I  Fn  dom  ( (
DIsoA `  K ) `  W )  ->  Fun  I )
64, 5syl 16 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Fun  I )
7 fvelrn 5947 . 2  |-  ( ( Fun  I  /\  X  e.  dom  I )  -> 
( I `  X
)  e.  ran  I
)
86, 7sylan 471 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
I `  X )  e.  ran  I )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   dom cdm 4947   ran crn 4948   Fun wfun 5519    Fn wfn 5520   ` cfv 5525   HLchlt 33318   LHypclh 33951   DIsoAcdia 34996   DIsoBcdib 35106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-dib 35107
This theorem is referenced by:  dibintclN  35135
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