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Theorem nrmsep3 23342
Description: In a normal space, given a closed set 𝐵 inside an open set 𝐴, there is an open set 𝑥 such that 𝐵𝑥 ⊆ cls(𝑥) ⊆ 𝐴. (Contributed by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
nrmsep3 ((𝐽 ∈ Nrm ∧ (𝐴𝐽𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴)) → ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 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.
StepHypRef Expression
1 isnrm 23322 . . . . 5 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑦𝐽𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑦)∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦)))
2 pweq 4620 . . . . . . . 8 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
32ineq2d 4212 . . . . . . 7 (𝑦 = 𝐴 → ((Clsd‘𝐽) ∩ 𝒫 𝑦) = ((Clsd‘𝐽) ∩ 𝒫 𝐴))
4 sseq2 4005 . . . . . . . . 9 (𝑦 = 𝐴 → (((cls‘𝐽)‘𝑥) ⊆ 𝑦 ↔ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))
54anbi2d 628 . . . . . . . 8 (𝑦 = 𝐴 → ((𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
65rexbidv 3168 . . . . . . 7 (𝑦 = 𝐴 → (∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
73, 6raleqbidv 3329 . . . . . 6 (𝑦 = 𝐴 → (∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑦)∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴)∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
87rspccv 3604 . . . . 5 (∀𝑦𝐽𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑦)∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) → (𝐴𝐽 → ∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴)∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
91, 8simplbiim 503 . . . 4 (𝐽 ∈ Nrm → (𝐴𝐽 → ∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴)∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
10 elin 3962 . . . . . 6 (𝐵 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴) ↔ (𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ 𝒫 𝐴))
11 elpwg 4609 . . . . . . 7 (𝐵 ∈ (Clsd‘𝐽) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
1211pm5.32i 573 . . . . . 6 ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ 𝒫 𝐴) ↔ (𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴))
1310, 12bitri 274 . . . . 5 (𝐵 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴) ↔ (𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴))
14 cleq1lem 14982 . . . . . . 7 (𝑧 = 𝐵 → ((𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) ↔ (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
1514rexbidv 3168 . . . . . 6 (𝑧 = 𝐵 → (∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) ↔ ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
1615rspccv 3604 . . . . 5 (∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴)∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) → (𝐵 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴) → ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
1713, 16biimtrrid 242 . . . 4 (∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴)∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) → ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
189, 17syl6 35 . . 3 (𝐽 ∈ Nrm → (𝐴𝐽 → ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))))
1918exp4a 430 . 2 (𝐽 ∈ Nrm → (𝐴𝐽 → (𝐵 ∈ (Clsd‘𝐽) → (𝐵𝐴 → ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))))
20193imp2 1346 1 ((𝐽 ∈ Nrm ∧ (𝐴𝐽𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴)) → ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  wrex 3059  cin 3945  wss 3946  𝒫 cpw 4606  cfv 6553  Topctop 22878  Clsdccld 23003  clsccl 23005  Nrmcnrm 23297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-nrm 23304
This theorem is referenced by:  nrmsep2  23343  kqnrmlem1  23730  kqnrmlem2  23731  nrmr0reg  23736  nrmhmph  23781
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