Theorem nrmsep3 21159

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 21139	. . . . 5 |- ( 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		pweq 4161	. . . . . . . 8 |- ( y = A -> ~P y = ~P A )
3	2	ineq2d 3814	. . . . . . 7 |- ( y = A -> ( ( Clsd ` J ) i^i ~P y ) = ( ( Clsd ` J ) i^i ~P A ) )
4		sseq2 3627	. . . . . . . . 9 |- ( y = A -> ( ( ( cls ` J ) ` x ) C_ y <-> ( ( cls ` J ) ` x ) C_ A ) )
5	4	anbi2d 740	. . . . . . . 8 |- ( y = A -> ( ( z C_ x /\ ( ( cls ` J ) ` x ) C_ y ) <-> ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
6	5	rexbidv 3052	. . . . . . 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 ) ) )
7	3, 6	raleqbidv 3152	. . . . . 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 ) ) )
8	7	rspccv 3306	. . . . 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 ) ) )
9	1, 8	simplbiim 659	. . . 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 ) ) )
10		elin 3796	. . . . . 6 |- ( B e. ( ( Clsd ` J ) i^i ~P A ) <-> ( B e. ( Clsd ` J ) /\ B e. ~P A ) )
11		elpwg 4166	. . . . . . 7 |- ( B e. ( Clsd ` J ) -> ( B e. ~P A <-> B C_ A ) )
12	11	pm5.32i 669	. . . . . 6 |- ( ( B e. ( Clsd ` J ) /\ B e. ~P A ) <-> ( B e. ( Clsd ` J ) /\ B C_ A ) )
13	10, 12	bitri 264	. . . . 5 |- ( B e. ( ( Clsd ` J ) i^i ~P A ) <-> ( B e. ( Clsd ` J ) /\ B C_ A ) )
14		cleq1lem 13721	. . . . . . 7 |- ( z = B -> ( ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) <-> ( B C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
15	14	rexbidv 3052	. . . . . 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 ) ) )
16	15	rspccv 3306	. . . . 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 ) ) )
17	13, 16	syl5bir 233	. . . 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 ) ) )
18	9, 17	syl6 35	. . 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 ) ) ) )
19	18	exp4a 633	. 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 ) ) ) ) )
20	19	3imp2 1282	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   ` cfv 5888   Topctop 20698   Clsdccld 20820   clsccl 20822   Nrmcnrm 21114

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-iota 5851  df-fv 5896  df-nrm 21121

This theorem is referenced by:  nrmsep2  21160  kqnrmlem1  21546  kqnrmlem2  21547  nrmr0reg  21552  nrmhmph  21597