Theorem neicvgbex 38410

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 o. ( D o. G ) )
neicvgbex.r	|- ( ph -> N H M )

Assertion

Ref	Expression
neicvgbex	|- ( ph -> B e. _V )

Proof of Theorem neicvgbex

Step	Hyp	Ref	Expression
1		neicvgbex.h	. . . . 5 |- H = ( F o. ( D o. G ) )
2		neicvgbex.d	. . . . . . 7 |- D = ( P ` B )
3	2	coeq1i 5281	. . . . . 6 |- ( D o. G ) = ( ( P ` B ) o. G )
4	3	coeq2i 5282	. . . . 5 |- ( F o. ( D o. G ) ) = ( F o. ( ( P ` B ) o. G ) )
5	1, 4	eqtri 2644	. . . 4 |- H = ( F o. ( ( P ` B ) o. G ) )
6	5	a1i 11	. . 3 |- ( ph -> H = ( F o. ( ( P ` B ) o. G ) ) )
7		neicvgbex.r	. . 3 |- ( ph -> N H M )
8	6, 7	breqdi 4668	. 2 |- ( ph -> N ( F o. ( ( P ` B ) o. G ) ) M )
9		brne0 4702	. 2 |- ( N ( F o. ( ( P ` B ) o. G ) ) M -> ( F o. ( ( P ` B ) o. G ) ) =/= (/) )
10		fvprc 6185	. . . . . . . . . . . . 13 |- ( -. B e. _V -> ( P ` B ) = (/) )
11	10	dmeqd 5326	. . . . . . . . . . . 12 |- ( -. B e. _V -> dom ( P ` B ) = dom (/) )
12		dm0 5339	. . . . . . . . . . . 12 |- dom (/) = (/)
13	11, 12	syl6eq 2672	. . . . . . . . . . 11 |- ( -. B e. _V -> dom ( P ` B ) = (/) )
14	13	ineq1d 3813	. . . . . . . . . 10 |- ( -. B e. _V -> ( dom ( P ` B ) i^i ran G ) = ( (/) i^i ran G ) )
15		incom 3805	. . . . . . . . . . 11 |- ( (/) i^i ran G ) = ( ran G i^i (/) )
16		in0 3968	. . . . . . . . . . 11 |- ( ran G i^i (/) ) = (/)
17	15, 16	eqtri 2644	. . . . . . . . . 10 |- ( (/) i^i ran G ) = (/)
18	14, 17	syl6eq 2672	. . . . . . . . 9 |- ( -. B e. _V -> ( dom ( P ` B ) i^i ran G ) = (/) )
19	18	coemptyd 13718	. . . . . . . 8 |- ( -. B e. _V -> ( ( P ` B ) o. G ) = (/) )
20	19	rneqd 5353	. . . . . . 7 |- ( -. B e. _V -> ran ( ( P ` B ) o. G ) = ran (/) )
21		rn0 5377	. . . . . . 7 |- ran (/) = (/)
22	20, 21	syl6eq 2672	. . . . . 6 |- ( -. B e. _V -> ran ( ( P ` B ) o. G ) = (/) )
23	22	ineq2d 3814	. . . . 5 |- ( -. B e. _V -> ( dom F i^i ran ( ( P ` B ) o. G ) ) = ( dom F i^i (/) ) )
24		in0 3968	. . . . 5 |- ( dom F i^i (/) ) = (/)
25	23, 24	syl6eq 2672	. . . 4 |- ( -. B e. _V -> ( dom F i^i ran ( ( P ` B ) o. G ) ) = (/) )
26	25	coemptyd 13718	. . 3 |- ( -. B e. _V -> ( F o. ( ( P ` B ) o. G ) ) = (/) )
27	26	necon1ai 2821	. 2 |- ( ( F o. ( ( P ` B ) o. G ) ) =/= (/) -> B e. _V )
28	8, 9, 27	3syl 18	1 |- ( ph -> B e. _V )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   i^i cin 3573   (/)c0 3915   class class class wbr 4653   dom cdm 5114   ran crn 5115   o. ccom 5118   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fv 5896

This theorem is referenced by:  neicvgrcomplex  38411  neicvgf1o  38412  neicvgnvo  38413  neicvgmex  38415  neicvgel1  38417