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Theorem clsneibex 44115
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 5871 . . . . 5 (𝐹𝐷) = (𝐹 ∘ (𝑃𝐵))
41, 3eqtri 2765 . . . 4 𝐻 = (𝐹 ∘ (𝑃𝐵))
54a1i 11 . . 3 (𝜑𝐻 = (𝐹 ∘ (𝑃𝐵)))
6 clsneibex.r . . 3 (𝜑𝐾𝐻𝑁)
75, 6breqdi 5158 . 2 (𝜑𝐾(𝐹 ∘ (𝑃𝐵))𝑁)
8 brne0 5193 . 2 (𝐾(𝐹 ∘ (𝑃𝐵))𝑁 → (𝐹 ∘ (𝑃𝐵)) ≠ ∅)
9 fvprc 6898 . . . . . . . 8 𝐵 ∈ V → (𝑃𝐵) = ∅)
109rneqd 5949 . . . . . . 7 𝐵 ∈ V → ran (𝑃𝐵) = ran ∅)
11 rn0 5936 . . . . . . 7 ran ∅ = ∅
1210, 11eqtrdi 2793 . . . . . 6 𝐵 ∈ V → ran (𝑃𝐵) = ∅)
1312ineq2d 4220 . . . . 5 𝐵 ∈ V → (dom 𝐹 ∩ ran (𝑃𝐵)) = (dom 𝐹 ∩ ∅))
14 in0 4395 . . . . 5 (dom 𝐹 ∩ ∅) = ∅
1513, 14eqtrdi 2793 . . . 4 𝐵 ∈ V → (dom 𝐹 ∩ ran (𝑃𝐵)) = ∅)
1615coemptyd 15018 . . 3 𝐵 ∈ V → (𝐹 ∘ (𝑃𝐵)) = ∅)
1716necon1ai 2968 . 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 1540  wcel 2108  wne 2940  Vcvv 3480  cin 3950  c0 4333   class class class wbr 5143  dom cdm 5685  ran crn 5686  ccom 5689  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fv 6569
This theorem is referenced by:  clsneircomplex  44116  clsneif1o  44117  clsneicnv  44118  clsneikex  44119  clsneinex  44120  clsneiel1  44121
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