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Theorem neicvgbex 40748
 Description: If (pseudo-)neighborhood and (pseudo-)convergent functions are related by the composite operator, 𝐻, then the base set exists. (Contributed by RP, 4-Jun-2021.)
Hypotheses
Ref Expression
neicvgbex.d 𝐷 = (𝑃𝐵)
neicvgbex.h 𝐻 = (𝐹 ∘ (𝐷𝐺))
neicvgbex.r (𝜑𝑁𝐻𝑀)
Assertion
Ref Expression
neicvgbex (𝜑𝐵 ∈ V)

Proof of Theorem neicvgbex
StepHypRef Expression
1 neicvgbex.h . . . . 5 𝐻 = (𝐹 ∘ (𝐷𝐺))
2 neicvgbex.d . . . . . . 7 𝐷 = (𝑃𝐵)
32coeq1i 5707 . . . . . 6 (𝐷𝐺) = ((𝑃𝐵) ∘ 𝐺)
43coeq2i 5708 . . . . 5 (𝐹 ∘ (𝐷𝐺)) = (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))
51, 4eqtri 2845 . . . 4 𝐻 = (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))
65a1i 11 . . 3 (𝜑𝐻 = (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺)))
7 neicvgbex.r . . 3 (𝜑𝑁𝐻𝑀)
86, 7breqdi 5057 . 2 (𝜑𝑁(𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))𝑀)
9 brne0 5092 . 2 (𝑁(𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))𝑀 → (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺)) ≠ ∅)
10 fvprc 6645 . . . . . . . . . . . . 13 𝐵 ∈ V → (𝑃𝐵) = ∅)
1110dmeqd 5751 . . . . . . . . . . . 12 𝐵 ∈ V → dom (𝑃𝐵) = dom ∅)
12 dm0 5767 . . . . . . . . . . . 12 dom ∅ = ∅
1311, 12syl6eq 2873 . . . . . . . . . . 11 𝐵 ∈ V → dom (𝑃𝐵) = ∅)
1413ineq1d 4162 . . . . . . . . . 10 𝐵 ∈ V → (dom (𝑃𝐵) ∩ ran 𝐺) = (∅ ∩ ran 𝐺))
15 0in 4319 . . . . . . . . . 10 (∅ ∩ ran 𝐺) = ∅
1614, 15syl6eq 2873 . . . . . . . . 9 𝐵 ∈ V → (dom (𝑃𝐵) ∩ ran 𝐺) = ∅)
1716coemptyd 14330 . . . . . . . 8 𝐵 ∈ V → ((𝑃𝐵) ∘ 𝐺) = ∅)
1817rneqd 5785 . . . . . . 7 𝐵 ∈ V → ran ((𝑃𝐵) ∘ 𝐺) = ran ∅)
19 rn0 5773 . . . . . . 7 ran ∅ = ∅
2018, 19syl6eq 2873 . . . . . 6 𝐵 ∈ V → ran ((𝑃𝐵) ∘ 𝐺) = ∅)
2120ineq2d 4163 . . . . 5 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃𝐵) ∘ 𝐺)) = (dom 𝐹 ∩ ∅))
22 in0 4317 . . . . 5 (dom 𝐹 ∩ ∅) = ∅
2321, 22syl6eq 2873 . . . 4 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃𝐵) ∘ 𝐺)) = ∅)
2423coemptyd 14330 . . 3 𝐵 ∈ V → (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺)) = ∅)
2524necon1ai 3038 . 2 ((𝐹 ∘ ((𝑃𝐵) ∘ 𝐺)) ≠ ∅ → 𝐵 ∈ V)
268, 9, 253syl 18 1 (𝜑𝐵 ∈ V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2114   ≠ wne 3011  Vcvv 3469   ∩ cin 3907  ∅c0 4265   class class class wbr 5042  dom cdm 5532  ran crn 5533   ∘ ccom 5536  ‘cfv 6334 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-iota 6293  df-fv 6342 This theorem is referenced by:  neicvgrcomplex  40749  neicvgf1o  40750  neicvgnvo  40751  neicvgmex  40753  neicvgel1  40755
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