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Theorem dibclN 31352
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 2283 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
3 dibcl.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
41, 2, 3dibfna 31344 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  dom  (
( DIsoA `  K ) `  W ) )
5 fnfun 5341 . . 3  |-  ( I  Fn  dom  ( (
DIsoA `  K ) `  W )  ->  Fun  I )
64, 5syl 15 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Fun  I )
7 fvelrn 5661 . 2  |-  ( ( Fun  I  /\  X  e.  dom  I )  -> 
( I `  X
)  e.  ran  I
)
86, 7sylan 457 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 358    = wceq 1623    e. wcel 1684   dom cdm 4689   ran crn 4690   Fun wfun 5249    Fn wfn 5250   ` cfv 5255   HLchlt 29540   LHypclh 30173   DIsoAcdia 31218   DIsoBcdib 31328
This theorem is referenced by:  dibintclN  31357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-dib 31329
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