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Theorem dibclN 31649
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 2408 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
3 dibcl.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
41, 2, 3dibfna 31641 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  dom  (
( DIsoA `  K ) `  W ) )
5 fnfun 5505 . . 3  |-  ( I  Fn  dom  ( (
DIsoA `  K ) `  W )  ->  Fun  I )
64, 5syl 16 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Fun  I )
7 fvelrn 5829 . 2  |-  ( ( Fun  I  /\  X  e.  dom  I )  -> 
( I `  X
)  e.  ran  I
)
86, 7sylan 458 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
I `  X )  e.  ran  I )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   dom cdm 4841   ran crn 4842   Fun wfun 5411    Fn wfn 5412   ` cfv 5417   HLchlt 29837   LHypclh 30470   DIsoAcdia 31515   DIsoBcdib 31625
This theorem is referenced by:  dibintclN  31654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-dib 31626
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