Theorem clsneibex 38400

Description: If (pseudo-)closure and (pseudo-)neighborhood functions are related by the composite operator, H, then the base set exists. (Contributed by RP, 4-Jun-2021.)

Hypotheses

Ref	Expression
clsneibex.d	|- D = ( P ` B )
clsneibex.h	|- H = ( F o. D )
clsneibex.r	|- ( ph -> K H N )

Assertion

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

Proof of Theorem clsneibex

Step	Hyp	Ref	Expression
1		clsneibex.h	. . . . 5 |- H = ( F o. D )
2		clsneibex.d	. . . . . 6 |- D = ( P ` B )
3	2	coeq2i 5282	. . . . 5 |- ( F o. D ) = ( F o. ( P ` B ) )
4	1, 3	eqtri 2644	. . . 4 |- H = ( F o. ( P ` B ) )
5	4	a1i 11	. . 3 |- ( ph -> H = ( F o. ( P ` B ) ) )
6		clsneibex.r	. . 3 |- ( ph -> K H N )
7	5, 6	breqdi 4668	. 2 |- ( ph -> K ( F o. ( P ` B ) ) N )
8		brne0 4702	. 2 |- ( K ( F o. ( P ` B ) ) N -> ( F o. ( P ` B ) ) =/= (/) )
9		fvprc 6185	. . . . . . . 8 |- ( -. B e. _V -> ( P ` B ) = (/) )
10	9	rneqd 5353	. . . . . . 7 |- ( -. B e. _V -> ran ( P ` B ) = ran (/) )
11		rn0 5377	. . . . . . 7 |- ran (/) = (/)
12	10, 11	syl6eq 2672	. . . . . 6 |- ( -. B e. _V -> ran ( P ` B ) = (/) )
13	12	ineq2d 3814	. . . . 5 |- ( -. B e. _V -> ( dom F i^i ran ( P ` B ) ) = ( dom F i^i (/) ) )
14		in0 3968	. . . . 5 |- ( dom F i^i (/) ) = (/)
15	13, 14	syl6eq 2672	. . . 4 |- ( -. B e. _V -> ( dom F i^i ran ( P ` B ) ) = (/) )
16	15	coemptyd 13718	. . 3 |- ( -. B e. _V -> ( F o. ( P ` B ) ) = (/) )
17	16	necon1ai 2821	. 2 |- ( ( F o. ( P ` B ) ) =/= (/) -> B e. _V )
18	7, 8, 17	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:  clsneircomplex  38401  clsneif1o  38402  clsneicnv  38403  clsneikex  38404  clsneinex  38405  clsneiel1  38406