Theorem neicvgbex 44468

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 5816	. . . . . 6 ⊢ (𝐷 ∘ 𝐺) = ((𝑃‘𝐵) ∘ 𝐺)
4	3	coeq2i 5817	. . . . 5 ⊢ (𝐹 ∘ (𝐷 ∘ 𝐺)) = (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))
5	1, 4	eqtri 2760	. . . 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 5115	. 2 ⊢ (𝜑 → 𝑁(𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))𝑀)
9		brne0 5150	. 2 ⊢ (𝑁(𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))𝑀 → (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) ≠ ∅)
10		fvprc 6834	. . . . . . . . . . . . 13 ⊢ (¬ 𝐵 ∈ V → (𝑃‘𝐵) = ∅)
11	10	dmeqd 5862	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 ∈ V → dom (𝑃‘𝐵) = dom ∅)
12		dm0 5877	. . . . . . . . . . . 12 ⊢ dom ∅ = ∅
13	11, 12	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (¬ 𝐵 ∈ V → dom (𝑃‘𝐵) = ∅)
14	13	ineq1d 4173	. . . . . . . . . 10 ⊢ (¬ 𝐵 ∈ V → (dom (𝑃‘𝐵) ∩ ran 𝐺) = (∅ ∩ ran 𝐺))
15		0in 4351	. . . . . . . . . 10 ⊢ (∅ ∩ ran 𝐺) = ∅
16	14, 15	eqtrdi 2788	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ V → (dom (𝑃‘𝐵) ∩ ran 𝐺) = ∅)
17	16	coemptyd 14914	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → ((𝑃‘𝐵) ∘ 𝐺) = ∅)
18	17	rneqd 5895	. . . . . . 7 ⊢ (¬ 𝐵 ∈ V → ran ((𝑃‘𝐵) ∘ 𝐺) = ran ∅)
19		rn0 5883	. . . . . . 7 ⊢ ran ∅ = ∅
20	18, 19	eqtrdi 2788	. . . . . 6 ⊢ (¬ 𝐵 ∈ V → ran ((𝑃‘𝐵) ∘ 𝐺) = ∅)
21	20	ineq2d 4174	. . . . 5 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃‘𝐵) ∘ 𝐺)) = (dom 𝐹 ∩ ∅))
22		in0 4349	. . . . 5 ⊢ (dom 𝐹 ∩ ∅) = ∅
23	21, 22	eqtrdi 2788	. . . 4 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃‘𝐵) ∘ 𝐺)) = ∅)
24	23	coemptyd 14914	. . 3 ⊢ (¬ 𝐵 ∈ V → (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) = ∅)
25	24	necon1ai 2960	. 2 ⊢ ((𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) ≠ ∅ → 𝐵 ∈ V)
26	8, 9, 25	3syl 18	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442   ∩ cin 3902  ∅c0 4287   class class class wbr 5100  dom cdm 5632  ran crn 5633   ∘ ccom 5636  ‘cfv 6500

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6456  df-fv 6508

This theorem is referenced by:  neicvgrcomplex  44469  neicvgf1o  44470  neicvgnvo  44471  neicvgmex  44473  neicvgel1  44475