MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nrmsep3 Structured version   Visualization version   GIF version

Theorem nrmsep3 22534
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 22514 . . . . 5 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑦𝐽𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑦)∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦)))
2 pweq 4552 . . . . . . . 8 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
32ineq2d 4149 . . . . . . 7 (𝑦 = 𝐴 → ((Clsd‘𝐽) ∩ 𝒫 𝑦) = ((Clsd‘𝐽) ∩ 𝒫 𝐴))
4 sseq2 3949 . . . . . . . . 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 3169 . . . . . . 7 (𝑦 = 𝐴 → (∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
73, 6raleqbidv 3338 . . . . . 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 3560 . . . . 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 504 . . . 4 (𝐽 ∈ Nrm → (𝐴𝐽 → ∀𝑧 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴)∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
10 elin 3905 . . . . . 6 (𝐵 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝐴) ↔ (𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ 𝒫 𝐴))
11 elpwg 4539 . . . . . . 7 (𝐵 ∈ (Clsd‘𝐽) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
1211pm5.32i 574 . . . . . 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 14721 . . . . . . 7 (𝑧 = 𝐵 → ((𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) ↔ (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
1514rexbidv 3169 . . . . . 6 (𝑧 = 𝐵 → (∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴) ↔ ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))
1615rspccv 3560 . . . . 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 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 431 . 2 (𝐽 ∈ Nrm → (𝐴𝐽 → (𝐵 ∈ (Clsd‘𝐽) → (𝐵𝐴 → ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴)))))
20193imp2 1347 1 ((𝐽 ∈ Nrm ∧ (𝐴𝐽𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴)) → ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1537  wcel 2101  wral 3059  wrex 3068  cin 3888  wss 3889  𝒫 cpw 4536  cfv 6447  Topctop 22070  Clsdccld 22195  clsccl 22197  Nrmcnrm 22489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-iota 6399  df-fv 6455  df-nrm 22496
This theorem is referenced by:  nrmsep2  22535  kqnrmlem1  22922  kqnrmlem2  22923  nrmr0reg  22928  nrmhmph  22973
  Copyright terms: Public domain W3C validator