Theorem regsep 21138

Description: In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.)

Assertion

Ref	Expression
regsep	|- ( ( J e. Reg /\ U e. J /\ A e. U ) -> E. x e. J ( A e. x /\ ( ( cls ` J ) ` x ) C_ U ) )

Distinct variable groups:   x, A   x, J   x, U

Proof of Theorem regsep

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isreg 21136	. . . 4 |- ( J e. Reg <-> ( J e. Top /\ A. y e. J A. z e. y E. x e. J ( z e. x /\ ( ( cls ` J ) ` x ) C_ y ) ) )
2		sseq2 3627	. . . . . . . 8 |- ( y = U -> ( ( ( cls ` J ) ` x ) C_ y <-> ( ( cls ` J ) ` x ) C_ U ) )
3	2	anbi2d 740	. . . . . . 7 |- ( y = U -> ( ( z e. x /\ ( ( cls ` J ) ` x ) C_ y ) <-> ( z e. x /\ ( ( cls ` J ) ` x ) C_ U ) ) )
4	3	rexbidv 3052	. . . . . 6 |- ( y = U -> ( E. x e. J ( z e. x /\ ( ( cls ` J ) ` x ) C_ y ) <-> E. x e. J ( z e. x /\ ( ( cls ` J ) ` x ) C_ U ) ) )
5	4	raleqbi1dv 3146	. . . . 5 |- ( y = U -> ( A. z e. y E. x e. J ( z e. x /\ ( ( cls ` J ) ` x ) C_ y ) <-> A. z e. U E. x e. J ( z e. x /\ ( ( cls ` J ) ` x ) C_ U ) ) )
6	5	rspccv 3306	. . . 4 |- ( A. y e. J A. z e. y E. x e. J ( z e. x /\ ( ( cls ` J ) ` x ) C_ y ) -> ( U e. J -> A. z e. U E. x e. J ( z e. x /\ ( ( cls ` J ) ` x ) C_ U ) ) )
7	1, 6	simplbiim 659	. . 3 |- ( J e. Reg -> ( U e. J -> A. z e. U E. x e. J ( z e. x /\ ( ( cls ` J ) ` x ) C_ U ) ) )
8		eleq1 2689	. . . . . 6 |- ( z = A -> ( z e. x <-> A e. x ) )
9	8	anbi1d 741	. . . . 5 |- ( z = A -> ( ( z e. x /\ ( ( cls ` J ) ` x ) C_ U ) <-> ( A e. x /\ ( ( cls ` J ) ` x ) C_ U ) ) )
10	9	rexbidv 3052	. . . 4 |- ( z = A -> ( E. x e. J ( z e. x /\ ( ( cls ` J ) ` x ) C_ U ) <-> E. x e. J ( A e. x /\ ( ( cls ` J ) ` x ) C_ U ) ) )
11	10	rspccv 3306	. . 3 |- ( A. z e. U E. x e. J ( z e. x /\ ( ( cls ` J ) ` x ) C_ U ) -> ( A e. U -> E. x e. J ( A e. x /\ ( ( cls ` J ) ` x ) C_ U ) ) )
12	7, 11	syl6 35	. 2 |- ( J e. Reg -> ( U e. J -> ( A e. U -> E. x e. J ( A e. x /\ ( ( cls ` J ) ` x ) C_ U ) ) ) )
13	12	3imp 1256	1 |- ( ( J e. Reg /\ U e. J /\ A e. U ) -> E. x e. J ( A e. x /\ ( ( cls ` J ) ` x ) C_ U ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   ` cfv 5888   Topctop 20698   clsccl 20822   Regcreg 21113

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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-reg 21120

This theorem is referenced by:  regsep2  21180  regr1lem  21542  kqreglem1  21544  kqreglem2  21545  reghmph  21596  cnextcn  21871