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Theorem clsneibex 38220
 Description: If (pseudo-)closure and (pseudo-)neighborhood functions are related by the composite operator, 𝐻, then the base set exists. (Contributed by RP, 4-Jun-2021.)
Hypotheses
Ref Expression
clsneibex.d 𝐷 = (𝑃𝐵)
clsneibex.h 𝐻 = (𝐹𝐷)
clsneibex.r (𝜑𝐾𝐻𝑁)
Assertion
Ref Expression
clsneibex (𝜑𝐵 ∈ V)

Proof of Theorem clsneibex
StepHypRef Expression
1 clsneibex.h . . . . 5 𝐻 = (𝐹𝐷)
2 clsneibex.d . . . . . 6 𝐷 = (𝑃𝐵)
32coeq2i 5271 . . . . 5 (𝐹𝐷) = (𝐹 ∘ (𝑃𝐵))
41, 3eqtri 2642 . . . 4 𝐻 = (𝐹 ∘ (𝑃𝐵))
54a1i 11 . . 3 (𝜑𝐻 = (𝐹 ∘ (𝑃𝐵)))
6 clsneibex.r . . 3 (𝜑𝐾𝐻𝑁)
75, 6breqdi 4659 . 2 (𝜑𝐾(𝐹 ∘ (𝑃𝐵))𝑁)
8 brne0 4693 . 2 (𝐾(𝐹 ∘ (𝑃𝐵))𝑁 → (𝐹 ∘ (𝑃𝐵)) ≠ ∅)
9 fvprc 6172 . . . . . . . 8 𝐵 ∈ V → (𝑃𝐵) = ∅)
109rneqd 5342 . . . . . . 7 𝐵 ∈ V → ran (𝑃𝐵) = ran ∅)
11 rn0 5366 . . . . . . 7 ran ∅ = ∅
1210, 11syl6eq 2670 . . . . . 6 𝐵 ∈ V → ran (𝑃𝐵) = ∅)
1312ineq2d 3806 . . . . 5 𝐵 ∈ V → (dom 𝐹 ∩ ran (𝑃𝐵)) = (dom 𝐹 ∩ ∅))
14 in0 3959 . . . . 5 (dom 𝐹 ∩ ∅) = ∅
1513, 14syl6eq 2670 . . . 4 𝐵 ∈ V → (dom 𝐹 ∩ ran (𝑃𝐵)) = ∅)
1615coemptyd 13699 . . 3 𝐵 ∈ V → (𝐹 ∘ (𝑃𝐵)) = ∅)
1716necon1ai 2818 . 2 ((𝐹 ∘ (𝑃𝐵)) ≠ ∅ → 𝐵 ∈ V)
187, 8, 173syl 18 1 (𝜑𝐵 ∈ V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1481   ∈ wcel 1988   ≠ wne 2791  Vcvv 3195   ∩ cin 3566  ∅c0 3907   class class class wbr 4644  dom cdm 5104  ran crn 5105   ∘ ccom 5108  ‘cfv 5876 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-iota 5839  df-fv 5884 This theorem is referenced by:  clsneircomplex  38221  clsneif1o  38222  clsneicnv  38223  clsneikex  38224  clsneinex  38225  clsneiel1  38226
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