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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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