Theorem nrmsep3 20371

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 20351	. . . . . 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 466	. . . . 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 3954	. . . . . . . 8 |- ( y = A -> ~P y = ~P A )
4	3	ineq2d 3634	. . . . . . 7 |- ( y = A -> ( ( Clsd ` J ) i^i ~P y ) = ( ( Clsd ` J ) i^i ~P A ) )
5		sseq2 3454	. . . . . . . . 9 |- ( y = A -> ( ( ( cls ` J ) ` x ) C_ y <-> ( ( cls ` J ) ` x ) C_ A ) )
6	5	anbi2d 710	. . . . . . . 8 |- ( y = A -> ( ( z C_ x /\ ( ( cls ` J ) ` x ) C_ y ) <-> ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
7	6	rexbidv 2901	. . . . . . 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 3001	. . . . . 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 3147	. . . . 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 17	. . . 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 3617	. . . . . 6 |- ( B e. ( ( Clsd ` J ) i^i ~P A ) <-> ( B e. ( Clsd ` J ) /\ B e. ~P A ) )
12		elpwg 3959	. . . . . . 7 |- ( B e. ( Clsd ` J ) -> ( B e. ~P A <-> B C_ A ) )
13	12	pm5.32i 643	. . . . . 6 |- ( ( B e. ( Clsd ` J ) /\ B e. ~P A ) <-> ( B e. ( Clsd ` J ) /\ B C_ A ) )
14	11, 13	bitri 253	. . . . 5 |- ( B e. ( ( Clsd ` J ) i^i ~P A ) <-> ( B e. ( Clsd ` J ) /\ B C_ A ) )
15		sseq1 3453	. . . . . . . 8 |- ( z = B -> ( z C_ x <-> B C_ x ) )
16	15	anbi1d 711	. . . . . . 7 |- ( z = B -> ( ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) <-> ( B C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
17	16	rexbidv 2901	. . . . . 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 3147	. . . . 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 222	. . . 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 34	. . 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 611	. 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 1224	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 371   /\ w3a 985   = wceq 1444   e. wcel 1887   A.wral 2737   E.wrex 2738   i^i cin 3403   C_ wss 3404   ~Pcpw 3951   ` cfv 5582   Topctop 19917   Clsdccld 20031   clsccl 20033   Nrmcnrm 20326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-iota 5546  df-fv 5590  df-nrm 20333

This theorem is referenced by:  nrmsep2  20372  kqnrmlem1  20758  kqnrmlem2  20759  nrmr0reg  20764  nrmhmph  20809