Theorem nrmsep3 19983

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 19963	. . . . . 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 4018	. . . . . . . 8 |- ( y = A -> ~P y = ~P A )
4	3	ineq2d 3696	. . . . . . 7 |- ( y = A -> ( ( Clsd ` J ) i^i ~P y ) = ( ( Clsd ` J ) i^i ~P A ) )
5		sseq2 3521	. . . . . . . . 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 2968	. . . . . . 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 3068	. . . . . 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 3207	. . . . 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 3683	. . . . . 6 |- ( B e. ( ( Clsd ` J ) i^i ~P A ) <-> ( B e. ( Clsd ` J ) /\ B e. ~P A ) )
12		elpwg 4023	. . . . . . 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 3520	. . . . . . . 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 2968	. . . . . 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 3207	. . . . 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 1395   e. wcel 1819   A.wral 2807   E.wrex 2808   i^i cin 3470   C_ wss 3471   ~Pcpw 4015   ` cfv 5594   Topctop 19521   Clsdccld 19644   clsccl 19646   Nrmcnrm 19938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-nrm 19945

This theorem is referenced by:  nrmsep2  19984  kqnrmlem1  20370  kqnrmlem2  20371  nrmr0reg  20376  nrmhmph  20421