Theorem neicvgbex 41675

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 = (𝑃‘𝐵)
neicvgbex.h	⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))
neicvgbex.r	⊢ (𝜑 → 𝑁𝐻𝑀)

Assertion

Ref	Expression
neicvgbex	⊢ (𝜑 → 𝐵 ∈ V)

Proof of Theorem neicvgbex

Step	Hyp	Ref	Expression
1		neicvgbex.h	. . . . 5 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))
2		neicvgbex.d	. . . . . . 7 ⊢ 𝐷 = (𝑃‘𝐵)
3	2	coeq1i 5765	. . . . . 6 ⊢ (𝐷 ∘ 𝐺) = ((𝑃‘𝐵) ∘ 𝐺)
4	3	coeq2i 5766	. . . . 5 ⊢ (𝐹 ∘ (𝐷 ∘ 𝐺)) = (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))
5	1, 4	eqtri 2767	. . . 4 ⊢ 𝐻 = (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))
6	5	a1i 11	. . 3 ⊢ (𝜑 → 𝐻 = (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)))
7		neicvgbex.r	. . 3 ⊢ (𝜑 → 𝑁𝐻𝑀)
8	6, 7	breqdi 5093	. 2 ⊢ (𝜑 → 𝑁(𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))𝑀)
9		brne0 5128	. 2 ⊢ (𝑁(𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))𝑀 → (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) ≠ ∅)
10		fvprc 6760	. . . . . . . . . . . . 13 ⊢ (¬ 𝐵 ∈ V → (𝑃‘𝐵) = ∅)
11	10	dmeqd 5811	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 ∈ V → dom (𝑃‘𝐵) = dom ∅)
12		dm0 5826	. . . . . . . . . . . 12 ⊢ dom ∅ = ∅
13	11, 12	eqtrdi 2795	. . . . . . . . . . 11 ⊢ (¬ 𝐵 ∈ V → dom (𝑃‘𝐵) = ∅)
14	13	ineq1d 4150	. . . . . . . . . 10 ⊢ (¬ 𝐵 ∈ V → (dom (𝑃‘𝐵) ∩ ran 𝐺) = (∅ ∩ ran 𝐺))
15		0in 4332	. . . . . . . . . 10 ⊢ (∅ ∩ ran 𝐺) = ∅
16	14, 15	eqtrdi 2795	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ V → (dom (𝑃‘𝐵) ∩ ran 𝐺) = ∅)
17	16	coemptyd 14671	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → ((𝑃‘𝐵) ∘ 𝐺) = ∅)
18	17	rneqd 5844	. . . . . . 7 ⊢ (¬ 𝐵 ∈ V → ran ((𝑃‘𝐵) ∘ 𝐺) = ran ∅)
19		rn0 5832	. . . . . . 7 ⊢ ran ∅ = ∅
20	18, 19	eqtrdi 2795	. . . . . 6 ⊢ (¬ 𝐵 ∈ V → ran ((𝑃‘𝐵) ∘ 𝐺) = ∅)
21	20	ineq2d 4151	. . . . 5 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃‘𝐵) ∘ 𝐺)) = (dom 𝐹 ∩ ∅))
22		in0 4330	. . . . 5 ⊢ (dom 𝐹 ∩ ∅) = ∅
23	21, 22	eqtrdi 2795	. . . 4 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃‘𝐵) ∘ 𝐺)) = ∅)
24	23	coemptyd 14671	. . 3 ⊢ (¬ 𝐵 ∈ V → (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) = ∅)
25	24	necon1ai 2972	. 2 ⊢ ((𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) ≠ ∅ → 𝐵 ∈ V)
26	8, 9, 25	3syl 18	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2109   ≠ wne 2944  Vcvv 3430   ∩ cin 3890  ∅c0 4261   class class class wbr 5078  dom cdm 5588  ran crn 5589   ∘ ccom 5592  ‘cfv 6430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-iota 6388  df-fv 6438

This theorem is referenced by:  neicvgrcomplex  41676  neicvgf1o  41677  neicvgnvo  41678  neicvgmex  41680  neicvgel1  41682