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Theorem neicvgbex 44125
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 5870 . . . . . 6 (𝐷𝐺) = ((𝑃𝐵) ∘ 𝐺)
43coeq2i 5871 . . . . 5 (𝐹 ∘ (𝐷𝐺)) = (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))
51, 4eqtri 2765 . . . 4 𝐻 = (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))
65a1i 11 . . 3 (𝜑𝐻 = (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺)))
7 neicvgbex.r . . 3 (𝜑𝑁𝐻𝑀)
86, 7breqdi 5158 . 2 (𝜑𝑁(𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))𝑀)
9 brne0 5193 . 2 (𝑁(𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))𝑀 → (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺)) ≠ ∅)
10 fvprc 6898 . . . . . . . . . . . . 13 𝐵 ∈ V → (𝑃𝐵) = ∅)
1110dmeqd 5916 . . . . . . . . . . . 12 𝐵 ∈ V → dom (𝑃𝐵) = dom ∅)
12 dm0 5931 . . . . . . . . . . . 12 dom ∅ = ∅
1311, 12eqtrdi 2793 . . . . . . . . . . 11 𝐵 ∈ V → dom (𝑃𝐵) = ∅)
1413ineq1d 4219 . . . . . . . . . 10 𝐵 ∈ V → (dom (𝑃𝐵) ∩ ran 𝐺) = (∅ ∩ ran 𝐺))
15 0in 4397 . . . . . . . . . 10 (∅ ∩ ran 𝐺) = ∅
1614, 15eqtrdi 2793 . . . . . . . . 9 𝐵 ∈ V → (dom (𝑃𝐵) ∩ ran 𝐺) = ∅)
1716coemptyd 15018 . . . . . . . 8 𝐵 ∈ V → ((𝑃𝐵) ∘ 𝐺) = ∅)
1817rneqd 5949 . . . . . . 7 𝐵 ∈ V → ran ((𝑃𝐵) ∘ 𝐺) = ran ∅)
19 rn0 5936 . . . . . . 7 ran ∅ = ∅
2018, 19eqtrdi 2793 . . . . . 6 𝐵 ∈ V → ran ((𝑃𝐵) ∘ 𝐺) = ∅)
2120ineq2d 4220 . . . . 5 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃𝐵) ∘ 𝐺)) = (dom 𝐹 ∩ ∅))
22 in0 4395 . . . . 5 (dom 𝐹 ∩ ∅) = ∅
2321, 22eqtrdi 2793 . . . 4 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃𝐵) ∘ 𝐺)) = ∅)
2423coemptyd 15018 . . 3 𝐵 ∈ V → (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺)) = ∅)
2524necon1ai 2968 . 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 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:  neicvgrcomplex  44126  neicvgf1o  44127  neicvgnvo  44128  neicvgmex  44130  neicvgel1  44132
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