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Theorem neicvgbex 38936
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 5420 . . . . . 6 (𝐷𝐺) = ((𝑃𝐵) ∘ 𝐺)
43coeq2i 5421 . . . . 5 (𝐹 ∘ (𝐷𝐺)) = (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))
51, 4eqtri 2793 . . . 4 𝐻 = (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))
65a1i 11 . . 3 (𝜑𝐻 = (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺)))
7 neicvgbex.r . . 3 (𝜑𝑁𝐻𝑀)
86, 7breqdi 4801 . 2 (𝜑𝑁(𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))𝑀)
9 brne0 4836 . 2 (𝑁(𝐹 ∘ ((𝑃𝐵) ∘ 𝐺))𝑀 → (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺)) ≠ ∅)
10 fvprc 6326 . . . . . . . . . . . . 13 𝐵 ∈ V → (𝑃𝐵) = ∅)
1110dmeqd 5464 . . . . . . . . . . . 12 𝐵 ∈ V → dom (𝑃𝐵) = dom ∅)
12 dm0 5477 . . . . . . . . . . . 12 dom ∅ = ∅
1311, 12syl6eq 2821 . . . . . . . . . . 11 𝐵 ∈ V → dom (𝑃𝐵) = ∅)
1413ineq1d 3964 . . . . . . . . . 10 𝐵 ∈ V → (dom (𝑃𝐵) ∩ ran 𝐺) = (∅ ∩ ran 𝐺))
15 incom 3956 . . . . . . . . . . 11 (∅ ∩ ran 𝐺) = (ran 𝐺 ∩ ∅)
16 in0 4112 . . . . . . . . . . 11 (ran 𝐺 ∩ ∅) = ∅
1715, 16eqtri 2793 . . . . . . . . . 10 (∅ ∩ ran 𝐺) = ∅
1814, 17syl6eq 2821 . . . . . . . . 9 𝐵 ∈ V → (dom (𝑃𝐵) ∩ ran 𝐺) = ∅)
1918coemptyd 13928 . . . . . . . 8 𝐵 ∈ V → ((𝑃𝐵) ∘ 𝐺) = ∅)
2019rneqd 5491 . . . . . . 7 𝐵 ∈ V → ran ((𝑃𝐵) ∘ 𝐺) = ran ∅)
21 rn0 5515 . . . . . . 7 ran ∅ = ∅
2220, 21syl6eq 2821 . . . . . 6 𝐵 ∈ V → ran ((𝑃𝐵) ∘ 𝐺) = ∅)
2322ineq2d 3965 . . . . 5 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃𝐵) ∘ 𝐺)) = (dom 𝐹 ∩ ∅))
24 in0 4112 . . . . 5 (dom 𝐹 ∩ ∅) = ∅
2523, 24syl6eq 2821 . . . 4 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃𝐵) ∘ 𝐺)) = ∅)
2625coemptyd 13928 . . 3 𝐵 ∈ V → (𝐹 ∘ ((𝑃𝐵) ∘ 𝐺)) = ∅)
2726necon1ai 2970 . 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 1631  wcel 2145  wne 2943  Vcvv 3351  cin 3722  c0 4063   class class class wbr 4786  dom cdm 5249  ran crn 5250  ccom 5253  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fv 6039
This theorem is referenced by:  neicvgrcomplex  38937  neicvgf1o  38938  neicvgnvo  38939  neicvgmex  38941  neicvgel1  38943
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