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Theorem dicdmN 41186
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 ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑝𝐴 ∣ ¬ 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 41185 . 2 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑝𝐴 ∣ ¬ 𝑝 𝑊})
65fndmd 6673 1 ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑝𝐴 ∣ ¬ 𝑝 𝑊})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436   class class class wbr 5143  dom cdm 5685  cfv 6561  lecple 17304  Atomscatm 39264  LHypclh 39986  DIsoCcdic 41174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-dic 41175
This theorem is referenced by:  dicvalrelN  41187
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