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Theorem dicdmN 35132
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 35131 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { p  e.  A  |  -.  p  .<_  W } )
6 fndm 5605 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2797   class class class wbr 4387   dom cdm 4935    Fn wfn 5508   ` cfv 5513   lecple 14344   Atomscatm 33211   LHypclh 33931   DIsoCcdic 35120
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 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-dic 35121
This theorem is referenced by:  dicvalrelN  35133
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