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Theorem dibclN 34162
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 2402 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
3 dibcl.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
41, 2, 3dibfna 34154 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  dom  (
( DIsoA `  K ) `  W ) )
5 fnfun 5658 . . 3  |-  ( I  Fn  dom  ( (
DIsoA `  K ) `  W )  ->  Fun  I )
64, 5syl 17 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Fun  I )
7 fvelrn 6001 . 2  |-  ( ( Fun  I  /\  X  e.  dom  I )  -> 
( I `  X
)  e.  ran  I
)
86, 7sylan 469 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 367    = wceq 1405    e. wcel 1842   dom cdm 4822   ran crn 4823   Fun wfun 5562    Fn wfn 5563   ` cfv 5568   HLchlt 32348   LHypclh 32981   DIsoAcdia 34028   DIsoBcdib 34138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-dib 34139
This theorem is referenced by:  dibintclN  34167
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