Theorem neicvgbex 44074

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 5813	. . . . . 6 ⊢ (𝐷 ∘ 𝐺) = ((𝑃‘𝐵) ∘ 𝐺)
4	3	coeq2i 5814	. . . . 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 5117	. 2 ⊢ (𝜑 → 𝑁(𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))𝑀)
9		brne0 5152	. 2 ⊢ (𝑁(𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))𝑀 → (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) ≠ ∅)
10		fvprc 6832	. . . . . . . . . . . . 13 ⊢ (¬ 𝐵 ∈ V → (𝑃‘𝐵) = ∅)
11	10	dmeqd 5859	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 ∈ V → dom (𝑃‘𝐵) = dom ∅)
12		dm0 5874	. . . . . . . . . . . 12 ⊢ dom ∅ = ∅
13	11, 12	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (¬ 𝐵 ∈ V → dom (𝑃‘𝐵) = ∅)
14	13	ineq1d 4178	. . . . . . . . . 10 ⊢ (¬ 𝐵 ∈ V → (dom (𝑃‘𝐵) ∩ ran 𝐺) = (∅ ∩ ran 𝐺))
15		0in 4356	. . . . . . . . . 10 ⊢ (∅ ∩ ran 𝐺) = ∅
16	14, 15	eqtrdi 2780	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ V → (dom (𝑃‘𝐵) ∩ ran 𝐺) = ∅)
17	16	coemptyd 14921	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → ((𝑃‘𝐵) ∘ 𝐺) = ∅)
18	17	rneqd 5891	. . . . . . 7 ⊢ (¬ 𝐵 ∈ V → ran ((𝑃‘𝐵) ∘ 𝐺) = ran ∅)
19		rn0 5879	. . . . . . 7 ⊢ ran ∅ = ∅
20	18, 19	eqtrdi 2780	. . . . . 6 ⊢ (¬ 𝐵 ∈ V → ran ((𝑃‘𝐵) ∘ 𝐺) = ∅)
21	20	ineq2d 4179	. . . . 5 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃‘𝐵) ∘ 𝐺)) = (dom 𝐹 ∩ ∅))
22		in0 4354	. . . . 5 ⊢ (dom 𝐹 ∩ ∅) = ∅
23	21, 22	eqtrdi 2780	. . . 4 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃‘𝐵) ∘ 𝐺)) = ∅)
24	23	coemptyd 14921	. . 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 3444   ∩ cin 3910  ∅c0 4292   class class class wbr 5102  dom cdm 5631  ran crn 5632   ∘ ccom 5635  ‘cfv 6499

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 5246  ax-nul 5256  ax-pr 5382

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6452  df-fv 6507

This theorem is referenced by:  neicvgrcomplex  44075  neicvgf1o  44076  neicvgnvo  44077  neicvgmex  44079  neicvgel1  44081