Theorem gneispaceel2 38442

Description: Every neighborhood of a point in a generic neighborhood space contains that point. (Contributed by RP, 15-Apr-2021.)

Hypothesis

Ref	Expression
gneispace.a	|- A = { f | ( f : dom f --> ( ~P ( ~P dom f \ { (/) } ) \ { (/) } ) /\ A. p e. dom f A. n e. ( f ` p ) ( p e. n /\ A. s e. ~P dom f ( n C_ s -> s e. ( f ` p ) ) ) ) }

Assertion

Ref	Expression
gneispaceel2	|- ( ( F e. A /\ P e. dom F /\ N e. ( F ` P ) ) -> P e. N )

Distinct variable groups:   n, F, p, f   F, s, f   f, n, p   P, p, n   n, N

Allowed substitution hints:   A( f, n, s, p)   P( f, s)   N( f, s, p)

Proof of Theorem gneispaceel2

Step	Hyp	Ref	Expression
1		gneispace.a	. . . . 5 |- A = { f | ( f : dom f --> ( ~P ( ~P dom f \ { (/) } ) \ { (/) } ) /\ A. p e. dom f A. n e. ( f ` p ) ( p e. n /\ A. s e. ~P dom f ( n C_ s -> s e. ( f ` p ) ) ) ) }
2	1	gneispaceel 38441	. . . 4 |- ( F e. A -> A. p e. dom F A. n e. ( F ` p ) p e. n )
3		fveq2 6191	. . . . . 6 |- ( p = P -> ( F ` p ) = ( F ` P ) )
4		eleq1 2689	. . . . . 6 |- ( p = P -> ( p e. n <-> P e. n ) )
5	3, 4	raleqbidv 3152	. . . . 5 |- ( p = P -> ( A. n e. ( F ` p ) p e. n <-> A. n e. ( F ` P ) P e. n ) )
6	5	rspccv 3306	. . . 4 |- ( A. p e. dom F A. n e. ( F ` p ) p e. n -> ( P e. dom F -> A. n e. ( F ` P ) P e. n ) )
7	2, 6	syl 17	. . 3 |- ( F e. A -> ( P e. dom F -> A. n e. ( F ` P ) P e. n ) )
8		eleq2 2690	. . . 4 |- ( n = N -> ( P e. n <-> P e. N ) )
9	8	rspccv 3306	. . 3 |- ( A. n e. ( F ` P ) P e. n -> ( N e. ( F ` P ) -> P e. N ) )
10	7, 9	syl6 35	. 2 |- ( F e. A -> ( P e. dom F -> ( N e. ( F ` P ) -> P e. N ) ) )
11	10	3imp 1256	1 |- ( ( F e. A /\ P e. dom F /\ N e. ( F ` P ) ) -> P e. N )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   dom cdm 5114   -->wf 5884   ` 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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by: (None)