Theorem gneispace3 38431

Description: The predicate that F is a (generic) Seifert And Threlfall neighborhood space. (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
gneispace3	|- ( F e. V -> ( F e. A <-> ( ( Fun F /\ ran F C_ ( ~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 ) ) ) ) ) )

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

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

Proof of Theorem gneispace3

Step	Hyp	Ref	Expression
1		gneispace.a	. . 3 |- 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	gneispace2 38430	. 2 |- ( F e. V -> ( F e. A <-> ( 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 ) ) ) ) ) )
3		df-f 5892	. . . 4 |- ( F : dom F --> ( ~P ( ~P dom F \ { (/) } ) \ { (/) } ) <-> ( F Fn dom F /\ ran F C_ ( ~P ( ~P dom F \ { (/) } ) \ { (/) } ) ) )
4		funfn 5918	. . . . 5 |- ( Fun F <-> F Fn dom F )
5	4	anbi1i 731	. . . 4 |- ( ( Fun F /\ ran F C_ ( ~P ( ~P dom F \ { (/) } ) \ { (/) } ) ) <-> ( F Fn dom F /\ ran F C_ ( ~P ( ~P dom F \ { (/) } ) \ { (/) } ) ) )
6	3, 5	bitr4i 267	. . 3 |- ( F : dom F --> ( ~P ( ~P dom F \ { (/) } ) \ { (/) } ) <-> ( Fun F /\ ran F C_ ( ~P ( ~P dom F \ { (/) } ) \ { (/) } ) ) )
7	6	anbi1i 731	. 2 |- ( ( 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 ) ) ) ) <-> ( ( Fun F /\ ran F C_ ( ~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 ) ) ) ) )
8	2, 7	syl6bb 276	1 |- ( F e. V -> ( F e. A <-> ( ( Fun F /\ ran F C_ ( ~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 ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->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:  gneispace  38432