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Theorem dicdmN 31350
Description: Domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dicfn.l  |-  .<_  =  ( le `  K )
dicfn.a  |-  A  =  ( Atoms `  K )
dicfn.h  |-  H  =  ( LHyp `  K
)
dicfn.i  |-  I  =  ( ( DIsoC `  K
) `  W )
Assertion
Ref Expression
dicdmN  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  {
p  e.  A  |  -.  p  .<_  W }
)
Distinct variable groups:    .<_ , p    A, p    K, p    W, p
Allowed substitution hints:    H( p)    I( p)    V( p)

Proof of Theorem dicdmN
StepHypRef Expression
1 dicfn.l . . 3  |-  .<_  =  ( le `  K )
2 dicfn.a . . 3  |-  A  =  ( Atoms `  K )
3 dicfn.h . . 3  |-  H  =  ( LHyp `  K
)
4 dicfn.i . . 3  |-  I  =  ( ( DIsoC `  K
) `  W )
51, 2, 3, 4dicfnN 31349 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { p  e.  A  |  -.  p  .<_  W } )
6 fndm 5477 . 2  |-  ( I  Fn  { p  e.  A  |  -.  p  .<_  W }  ->  dom  I  =  { p  e.  A  |  -.  p  .<_  W } )
75, 6syl 16 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  {
p  e.  A  |  -.  p  .<_  W }
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2646   class class class wbr 4146   dom cdm 4811    Fn wfn 5382   ` cfv 5387   lecple 13456   Atomscatm 29429   LHypclh 30149   DIsoCcdic 31338
This theorem is referenced by:  dicvalrelN  31351
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-riota 6478  df-dic 31339
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