Theorem dicdmN 41512

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 41511	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊})
6	5	fndmd 6598	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3400   class class class wbr 5099  dom cdm 5625  ‘cfv 6493  lecple 17188  Atomscatm 39591  LHypclh 40312  DIsoCcdic 41500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-dic 41501

This theorem is referenced by:  dicvalrelN  41513