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Theorem nrmsep3 21670
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 21650 . . . . 5 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑦𝐽𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑦)∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦)))
2 pweq 4426 . . . . . . . 8 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
32ineq2d 4078 . . . . . . 7 (𝑦 = 𝐴 → ((Clsd‘𝐽) ∩ 𝒫 𝑦) = ((Clsd‘𝐽) ∩ 𝒫 𝐴))
4 sseq2 3885 . . . . . . . . 9 (𝑦 = 𝐴 → (((cls‘𝐽)‘𝑥) ⊆ 𝑦 ↔ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))
54anbi2d 619 . . . . . . . 8 (𝑦 = 𝐴 → ((𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
65rexbidv 3242 . . . . . . 7 (𝑦 = 𝐴 → (∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
73, 6raleqbidv 3341 . . . . . 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 3532 . . . . 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 497 . . . 4 (𝐽 ∈ Nrm → (𝐴𝐽 → ∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴)∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
10 elin 4059 . . . . . 6 (𝐵 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴) ↔ (𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ 𝒫 𝐴))
11 elpwg 4431 . . . . . . 7 (𝐵 ∈ (Clsd‘𝐽) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
1211pm5.32i 567 . . . . . 6 ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ 𝒫 𝐴) ↔ (𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴))
1310, 12bitri 267 . . . . 5 (𝐵 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴) ↔ (𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴))
14 cleq1lem 14206 . . . . . . 7 (𝑧 = 𝐵 → ((𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) ↔ (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
1514rexbidv 3242 . . . . . 6 (𝑧 = 𝐵 → (∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) ↔ ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
1615rspccv 3532 . . . . 5 (∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴)∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) → (𝐵 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴) → ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
1713, 16syl5bir 235 . . . 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 424 . 2 (𝐽 ∈ Nrm → (𝐴𝐽 → (𝐵 ∈ (Clsd‘𝐽) → (𝐵𝐴 → ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))))
20193imp2 1329 1 ((𝐽 ∈ Nrm ∧ (𝐴𝐽𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴)) → ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2050  wral 3088  wrex 3089  cin 3830  wss 3831  𝒫 cpw 4423  cfv 6190  Topctop 21208  Clsdccld 21331  clsccl 21333  Nrmcnrm 21625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-iota 6154  df-fv 6198  df-nrm 21632
This theorem is referenced by:  nrmsep2  21671  kqnrmlem1  22058  kqnrmlem2  22059  nrmr0reg  22064  nrmhmph  22109
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