Theorem neicvgbex 44085

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 5802	. . . . . 6 ⊢ (𝐷 ∘ 𝐺) = ((𝑃‘𝐵) ∘ 𝐺)
4	3	coeq2i 5803	. . . . 5 ⊢ (𝐹 ∘ (𝐷 ∘ 𝐺)) = (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))
5	1, 4	eqtri 2752	. . . 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 5107	. 2 ⊢ (𝜑 → 𝑁(𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))𝑀)
9		brne0 5142	. 2 ⊢ (𝑁(𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))𝑀 → (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) ≠ ∅)
10		fvprc 6814	. . . . . . . . . . . . 13 ⊢ (¬ 𝐵 ∈ V → (𝑃‘𝐵) = ∅)
11	10	dmeqd 5848	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 ∈ V → dom (𝑃‘𝐵) = dom ∅)
12		dm0 5863	. . . . . . . . . . . 12 ⊢ dom ∅ = ∅
13	11, 12	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (¬ 𝐵 ∈ V → dom (𝑃‘𝐵) = ∅)
14	13	ineq1d 4170	. . . . . . . . . 10 ⊢ (¬ 𝐵 ∈ V → (dom (𝑃‘𝐵) ∩ ran 𝐺) = (∅ ∩ ran 𝐺))
15		0in 4348	. . . . . . . . . 10 ⊢ (∅ ∩ ran 𝐺) = ∅
16	14, 15	eqtrdi 2780	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ V → (dom (𝑃‘𝐵) ∩ ran 𝐺) = ∅)
17	16	coemptyd 14886	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → ((𝑃‘𝐵) ∘ 𝐺) = ∅)
18	17	rneqd 5880	. . . . . . 7 ⊢ (¬ 𝐵 ∈ V → ran ((𝑃‘𝐵) ∘ 𝐺) = ran ∅)
19		rn0 5868	. . . . . . 7 ⊢ ran ∅ = ∅
20	18, 19	eqtrdi 2780	. . . . . 6 ⊢ (¬ 𝐵 ∈ V → ran ((𝑃‘𝐵) ∘ 𝐺) = ∅)
21	20	ineq2d 4171	. . . . 5 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃‘𝐵) ∘ 𝐺)) = (dom 𝐹 ∩ ∅))
22		in0 4346	. . . . 5 ⊢ (dom 𝐹 ∩ ∅) = ∅
23	21, 22	eqtrdi 2780	. . . 4 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃‘𝐵) ∘ 𝐺)) = ∅)
24	23	coemptyd 14886	. . 3 ⊢ (¬ 𝐵 ∈ V → (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) = ∅)
25	24	necon1ai 2952	. 2 ⊢ ((𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) ≠ ∅ → 𝐵 ∈ V)
26	8, 9, 25	3syl 18	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3436   ∩ cin 3902  ∅c0 4284   class class class wbr 5092  dom cdm 5619  ran crn 5620   ∘ ccom 5623  ‘cfv 6482

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-iota 6438  df-fv 6490

This theorem is referenced by:  neicvgrcomplex  44086  neicvgf1o  44087  neicvgnvo  44088  neicvgmex  44090  neicvgel1  44092