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Theorem dicdmN 36994
 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 36993 . 2 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑝𝐴 ∣ ¬ 𝑝 𝑊})
6 fndm 6129 . 2 (𝐼 Fn {𝑝𝐴 ∣ ¬ 𝑝 𝑊} → dom 𝐼 = {𝑝𝐴 ∣ ¬ 𝑝 𝑊})
75, 6syl 17 1 ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑝𝐴 ∣ ¬ 𝑝 𝑊})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  {crab 3065   class class class wbr 4787  dom cdm 5250   Fn wfn 6025  ‘cfv 6030  lecple 16156  Atomscatm 35072  LHypclh 35793  DIsoCcdic 36982 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-dic 36983 This theorem is referenced by:  dicvalrelN  36995
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