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Theorem neicvgbex 39366
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 5527 . . . . . 6 (𝐷𝐺) = ((𝑃𝐵) ∘ 𝐺)
43coeq2i 5528 . . . . 5 (𝐹 ∘ (𝐷𝐺)) = (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))
51, 4eqtri 2802 . . . 4 𝐻 = (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))
65a1i 11 . . 3 (𝜑𝐻 = (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺)))
7 neicvgbex.r . . 3 (𝜑𝑁𝐻𝑀)
86, 7breqdi 4901 . 2 (𝜑𝑁(𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))𝑀)
9 brne0 4936 . 2 (𝑁(𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))𝑀 → (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺)) ≠ ∅)
10 fvprc 6439 . . . . . . . . . . . . 13 𝐵 ∈ V → (𝑃𝐵) = ∅)
1110dmeqd 5571 . . . . . . . . . . . 12 𝐵 ∈ V → dom (𝑃𝐵) = dom ∅)
12 dm0 5584 . . . . . . . . . . . 12 dom ∅ = ∅
1311, 12syl6eq 2830 . . . . . . . . . . 11 𝐵 ∈ V → dom (𝑃𝐵) = ∅)
1413ineq1d 4036 . . . . . . . . . 10 𝐵 ∈ V → (dom (𝑃𝐵) ∩ ran 𝐺) = (∅ ∩ ran 𝐺))
15 incom 4028 . . . . . . . . . . 11 (∅ ∩ ran 𝐺) = (ran 𝐺 ∩ ∅)
16 in0 4194 . . . . . . . . . . 11 (ran 𝐺 ∩ ∅) = ∅
1715, 16eqtri 2802 . . . . . . . . . 10 (∅ ∩ ran 𝐺) = ∅
1814, 17syl6eq 2830 . . . . . . . . 9 𝐵 ∈ V → (dom (𝑃𝐵) ∩ ran 𝐺) = ∅)
1918coemptyd 14127 . . . . . . . 8 𝐵 ∈ V → ((𝑃𝐵) ∘ 𝐺) = ∅)
2019rneqd 5598 . . . . . . 7 𝐵 ∈ V → ran ((𝑃𝐵) ∘ 𝐺) = ran ∅)
21 rn0 5623 . . . . . . 7 ran ∅ = ∅
2220, 21syl6eq 2830 . . . . . 6 𝐵 ∈ V → ran ((𝑃𝐵) ∘ 𝐺) = ∅)
2322ineq2d 4037 . . . . 5 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃𝐵) ∘ 𝐺)) = (dom 𝐹 ∩ ∅))
24 in0 4194 . . . . 5 (dom 𝐹 ∩ ∅) = ∅
2523, 24syl6eq 2830 . . . 4 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃𝐵) ∘ 𝐺)) = ∅)
2625coemptyd 14127 . . 3 𝐵 ∈ V → (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺)) = ∅)
2726necon1ai 2996 . 2 ((𝐹 ∘ ((𝑃𝐵) ∘ 𝐺)) ≠ ∅ → 𝐵 ∈ V)
288, 9, 273syl 18 1 (𝜑𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1601  wcel 2107  wne 2969  Vcvv 3398  cin 3791  c0 4141   class class class wbr 4886  dom cdm 5355  ran crn 5356  ccom 5359  cfv 6135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-iota 6099  df-fv 6143
This theorem is referenced by:  neicvgrcomplex  39367  neicvgf1o  39368  neicvgnvo  39369  neicvgmex  39371  neicvgel1  39373
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