Theorem nrmsep3 20448

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 20428	. . . . . 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 471	. . . . 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 3945	. . . . . . . 8 |- ( y = A -> ~P y = ~P A )
4	3	ineq2d 3625	. . . . . . 7 |- ( y = A -> ( ( Clsd ` J ) i^i ~P y ) = ( ( Clsd ` J ) i^i ~P A ) )
5		sseq2 3440	. . . . . . . . 9 |- ( y = A -> ( ( ( cls ` J ) ` x ) C_ y <-> ( ( cls ` J ) ` x ) C_ A ) )
6	5	anbi2d 718	. . . . . . . 8 |- ( y = A -> ( ( z C_ x /\ ( ( cls ` J ) ` x ) C_ y ) <-> ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
7	6	rexbidv 2892	. . . . . . 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 2987	. . . . . 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 3133	. . . . 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 3608	. . . . . 6 |- ( B e. ( ( Clsd ` J ) i^i ~P A ) <-> ( B e. ( Clsd ` J ) /\ B e. ~P A ) )
12		elpwg 3950	. . . . . . 7 |- ( B e. ( Clsd ` J ) -> ( B e. ~P A <-> B C_ A ) )
13	12	pm5.32i 649	. . . . . 6 |- ( ( B e. ( Clsd ` J ) /\ B e. ~P A ) <-> ( B e. ( Clsd ` J ) /\ B C_ A ) )
14	11, 13	bitri 257	. . . . 5 |- ( B e. ( ( Clsd ` J ) i^i ~P A ) <-> ( B e. ( Clsd ` J ) /\ B C_ A ) )
15		sseq1 3439	. . . . . . . 8 |- ( z = B -> ( z C_ x <-> B C_ x ) )
16	15	anbi1d 719	. . . . . . 7 |- ( z = B -> ( ( z C_ x /\ ( ( cls ` J ) ` x ) C_ A ) <-> ( B C_ x /\ ( ( cls ` J ) ` x ) C_ A ) ) )
17	16	rexbidv 2892	. . . . . 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 3133	. . . . 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 226	. . . 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 617	. 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 1248	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 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   E.wrex 2757   i^i cin 3389   C_ wss 3390   ~Pcpw 3942   ` cfv 5589   Topctop 19994   Clsdccld 20108   clsccl 20110   Nrmcnrm 20403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-nrm 20410

This theorem is referenced by:  nrmsep2  20449  kqnrmlem1  20835  kqnrmlem2  20836  nrmr0reg  20841  nrmhmph  20886