Theorem neicvgbex 40815

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 5694	. . . . . 6 ⊢ (𝐷 ∘ 𝐺) = ((𝑃‘𝐵) ∘ 𝐺)
4	3	coeq2i 5695	. . . . 5 ⊢ (𝐹 ∘ (𝐷 ∘ 𝐺)) = (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))
5	1, 4	eqtri 2821	. . . 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 5045	. 2 ⊢ (𝜑 → 𝑁(𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))𝑀)
9		brne0 5080	. 2 ⊢ (𝑁(𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))𝑀 → (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) ≠ ∅)
10		fvprc 6638	. . . . . . . . . . . . 13 ⊢ (¬ 𝐵 ∈ V → (𝑃‘𝐵) = ∅)
11	10	dmeqd 5738	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 ∈ V → dom (𝑃‘𝐵) = dom ∅)
12		dm0 5754	. . . . . . . . . . . 12 ⊢ dom ∅ = ∅
13	11, 12	eqtrdi 2849	. . . . . . . . . . 11 ⊢ (¬ 𝐵 ∈ V → dom (𝑃‘𝐵) = ∅)
14	13	ineq1d 4138	. . . . . . . . . 10 ⊢ (¬ 𝐵 ∈ V → (dom (𝑃‘𝐵) ∩ ran 𝐺) = (∅ ∩ ran 𝐺))
15		0in 4301	. . . . . . . . . 10 ⊢ (∅ ∩ ran 𝐺) = ∅
16	14, 15	eqtrdi 2849	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ V → (dom (𝑃‘𝐵) ∩ ran 𝐺) = ∅)
17	16	coemptyd 14330	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → ((𝑃‘𝐵) ∘ 𝐺) = ∅)
18	17	rneqd 5772	. . . . . . 7 ⊢ (¬ 𝐵 ∈ V → ran ((𝑃‘𝐵) ∘ 𝐺) = ran ∅)
19		rn0 5760	. . . . . . 7 ⊢ ran ∅ = ∅
20	18, 19	eqtrdi 2849	. . . . . 6 ⊢ (¬ 𝐵 ∈ V → ran ((𝑃‘𝐵) ∘ 𝐺) = ∅)
21	20	ineq2d 4139	. . . . 5 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃‘𝐵) ∘ 𝐺)) = (dom 𝐹 ∩ ∅))
22		in0 4299	. . . . 5 ⊢ (dom 𝐹 ∩ ∅) = ∅
23	21, 22	eqtrdi 2849	. . . 4 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃‘𝐵) ∘ 𝐺)) = ∅)
24	23	coemptyd 14330	. . 3 ⊢ (¬ 𝐵 ∈ V → (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) = ∅)
25	24	necon1ai 3014	. 2 ⊢ ((𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) ≠ ∅ → 𝐵 ∈ V)
26	8, 9, 25	3syl 18	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441   ∩ cin 3880  ∅c0 4243   class class class wbr 5030  dom cdm 5519  ran crn 5520   ∘ ccom 5523  ‘cfv 6324

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fv 6332

This theorem is referenced by:  neicvgrcomplex  40816  neicvgf1o  40817  neicvgnvo  40818  neicvgmex  40820  neicvgel1  40822