Theorem dicdmN 39198

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 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑝 ∈ 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

Step	Hyp	Ref	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)
5	1, 2, 3, 4	dicfnN 39197	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊})
6	5	fndmd 6538	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068   class class class wbr 5074  dom cdm 5589  ‘cfv 6433  lecple 16969  Atomscatm 37277  LHypclh 37998  DIsoCcdic 39186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-dic 39187

This theorem is referenced by:  dicvalrelN  39199