Theorem nrmsep3 19662

Description: In a normal space, given a closed set B inside an open set A, there is an open set x such that B C_ x C_ cls ( x ) C_ A. (Contributed by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
nrmsep3	|- ( ( J e. Nrm /\ ( A e. J /\ B e. ( Clsd ` J ) /\ B C_ A ) ) -> E. x e. J ( B C_ x /\ ( ( cls ` J ) ` x ) C_ A ) )

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

Proof of Theorem nrmsep3

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

Step	Hyp	Ref	Expression
1		isnrm 19642	. . . . . 6 |- ( J e. Nrm <-> ( J e. Top /\ A. y e. J A. z e. ( ( Clsd ` J ) i^i ~P y ) E. x e. J ( z C_ x /\ ( ( cls ` J ) ` x ) C_ y ) ) )
2	1	simprbi 464	. . . . 5 |- ( J e. Nrm -> A. y e. J A. z e. ( ( Clsd ` J ) i^i ~P y ) E. x e. J ( z C_ x /\ ( ( cls ` J ) ` x ) C_ y ) )
3		pweq 4013	. . . . . . . 8 |- ( y = A -> ~P y = ~P A )
4	3	ineq2d 3700	. . . . . . 7 |- ( y = A -> ( ( Clsd ` J ) i^i ~P y ) = ( ( Clsd ` J ) i^i ~P A ) )
5		sseq2 3526	. . . . . . . . 9 |- ( y = A -> ( ( ( cls ` J ) ` x ) C_ y <-> ( ( cls ` J ) ` x ) C_ A ) )
6	5	anbi2d 703	. . . . . . . 8 |- ( y = A -> ( ( z C_ x /\ ( ( cls ` J ) ` x ) C_ y ) <-> ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
7	6	rexbidv 2973	. . . . . . 7 |- ( y = A -> ( E. x e. J ( z C_ x /\ ( ( cls ` J ) ` x ) C_ y ) <-> E. x e. J ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
8	4, 7	raleqbidv 3072	. . . . . 6 |- ( y = A -> ( A. z e. ( ( Clsd ` J ) i^i ~P y ) E. x e. J ( z C_ x /\ ( ( cls ` J ) ` x ) C_ y ) <-> A. z e. ( ( Clsd ` J ) i^i ~P A ) E. x e. J ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
9	8	rspccv 3211	. . . . 5 |- ( A. y e. J A. z e. ( ( Clsd ` J ) i^i ~P y ) E. x e. J ( z C_ x /\ ( ( cls ` J ) ` x ) C_ y ) -> ( A e. J -> A. z e. ( ( Clsd ` J ) i^i ~P A ) E. x e. J ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
10	2, 9	syl 16	. . . 4 |- ( J e. Nrm -> ( A e. J -> A. z e. ( ( Clsd ` J ) i^i ~P A ) E. x e. J ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
11		elin 3687	. . . . . 6 |- ( B e. ( ( Clsd ` J ) i^i ~P A ) <-> ( B e. ( Clsd ` J ) /\ B e. ~P A ) )
12		elpwg 4018	. . . . . . 7 |- ( B e. ( Clsd ` J ) -> ( B e. ~P A <-> B C_ A ) )
13	12	pm5.32i 637	. . . . . 6 |- ( ( B e. ( Clsd ` J ) /\ B e. ~P A ) <-> ( B e. ( Clsd ` J ) /\ B C_ A ) )
14	11, 13	bitri 249	. . . . 5 |- ( B e. ( ( Clsd ` J ) i^i ~P A ) <-> ( B e. ( Clsd ` J ) /\ B C_ A ) )
15		sseq1 3525	. . . . . . . 8 |- ( z = B -> ( z C_ x <-> B C_ x ) )
16	15	anbi1d 704	. . . . . . 7 |- ( z = B -> ( ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) <-> ( B C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
17	16	rexbidv 2973	. . . . . 6 |- ( z = B -> ( E. x e. J ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) <-> E. x e. J ( B C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
18	17	rspccv 3211	. . . . 5 |- ( A. z e. ( ( Clsd ` J ) i^i ~P A ) E. x e. J ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) -> ( B e. ( ( Clsd ` J ) i^i ~P A ) -> E. x e. J ( B C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
19	14, 18	syl5bir 218	. . . 4 |- ( A. z e. ( ( Clsd ` J ) i^i ~P A ) E. x e. J ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) -> ( ( B e. ( Clsd ` J ) /\ B C_ A ) -> E. x e. J ( B C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
20	10, 19	syl6 33	. . 3 |- ( J e. Nrm -> ( A e. J -> ( ( B e. ( Clsd ` J ) /\ B C_ A ) -> E. x e. J ( B C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) ) )
21	20	exp4a 606	. 2 |- ( J e. Nrm -> ( A e. J -> ( B e. ( Clsd ` J ) -> ( B C_ A -> E. x e. J ( B C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) ) ) )
22	21	3imp2 1211	1 |- ( ( J e. Nrm /\ ( A e. J /\ B e. ( Clsd ` J ) /\ B C_ A ) ) -> E. x e. J ( B C_ x /\ ( ( cls ` J ) ` x ) C_ A ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   A.wral 2814   E.wrex 2815   i^i cin 3475   C_ wss 3476   ~Pcpw 4010   ` cfv 5588   Topctop 19201   Clsdccld 19323   clsccl 19325   Nrmcnrm 19617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-nrm 19624

This theorem is referenced by:  nrmsep2  19663  kqnrmlem1  20071  kqnrmlem2  20072  nrmr0reg  20077  nrmhmph  20122