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Theorem clsneibex 38920
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 5438 . . . . 5 (𝐹𝐷) = (𝐹 ∘ (𝑃𝐵))
41, 3eqtri 2782 . . . 4 𝐻 = (𝐹 ∘ (𝑃𝐵))
54a1i 11 . . 3 (𝜑𝐻 = (𝐹 ∘ (𝑃𝐵)))
6 clsneibex.r . . 3 (𝜑𝐾𝐻𝑁)
75, 6breqdi 4819 . 2 (𝜑𝐾(𝐹 ∘ (𝑃𝐵))𝑁)
8 brne0 4854 . 2 (𝐾(𝐹 ∘ (𝑃𝐵))𝑁 → (𝐹 ∘ (𝑃𝐵)) ≠ ∅)
9 fvprc 6347 . . . . . . . 8 𝐵 ∈ V → (𝑃𝐵) = ∅)
109rneqd 5508 . . . . . . 7 𝐵 ∈ V → ran (𝑃𝐵) = ran ∅)
11 rn0 5532 . . . . . . 7 ran ∅ = ∅
1210, 11syl6eq 2810 . . . . . 6 𝐵 ∈ V → ran (𝑃𝐵) = ∅)
1312ineq2d 3957 . . . . 5 𝐵 ∈ V → (dom 𝐹 ∩ ran (𝑃𝐵)) = (dom 𝐹 ∩ ∅))
14 in0 4111 . . . . 5 (dom 𝐹 ∩ ∅) = ∅
1513, 14syl6eq 2810 . . . 4 𝐵 ∈ V → (dom 𝐹 ∩ ran (𝑃𝐵)) = ∅)
1615coemptyd 13939 . . 3 𝐵 ∈ V → (𝐹 ∘ (𝑃𝐵)) = ∅)
1716necon1ai 2959 . 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 1632  wcel 2139  wne 2932  Vcvv 3340  cin 3714  c0 4058   class class class wbr 4804  dom cdm 5266  ran crn 5267  ccom 5270  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fv 6057
This theorem is referenced by:  clsneircomplex  38921  clsneif1o  38922  clsneicnv  38923  clsneikex  38924  clsneinex  38925  clsneiel1  38926
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