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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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