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Theorem regsep 23251
Description: In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
regsep ((𝐽 ∈ Reg ∧ 𝑈𝐽𝐴𝑈) → ∃𝑥𝐽 (𝐴𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑈

Proof of Theorem regsep
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isreg 23249 . . . 4 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑦𝐽𝑧𝑦𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦)))
2 sseq2 4006 . . . . . . . 8 (𝑦 = 𝑈 → (((cls‘𝐽)‘𝑥) ⊆ 𝑦 ↔ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))
32anbi2d 629 . . . . . . 7 (𝑦 = 𝑈 → ((𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
43rexbidv 3175 . . . . . 6 (𝑦 = 𝑈 → (∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
54raleqbi1dv 3330 . . . . 5 (𝑦 = 𝑈 → (∀𝑧𝑦𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∀𝑧𝑈𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
65rspccv 3606 . . . 4 (∀𝑦𝐽𝑧𝑦𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) → (𝑈𝐽 → ∀𝑧𝑈𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
71, 6simplbiim 504 . . 3 (𝐽 ∈ Reg → (𝑈𝐽 → ∀𝑧𝑈𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
8 eleq1 2817 . . . . . 6 (𝑧 = 𝐴 → (𝑧𝑥𝐴𝑥))
98anbi1d 630 . . . . 5 (𝑧 = 𝐴 → ((𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈) ↔ (𝐴𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
109rexbidv 3175 . . . 4 (𝑧 = 𝐴 → (∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈) ↔ ∃𝑥𝐽 (𝐴𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
1110rspccv 3606 . . 3 (∀𝑧𝑈𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈) → (𝐴𝑈 → ∃𝑥𝐽 (𝐴𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
127, 11syl6 35 . 2 (𝐽 ∈ Reg → (𝑈𝐽 → (𝐴𝑈 → ∃𝑥𝐽 (𝐴𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))))
13123imp 1109 1 ((𝐽 ∈ Reg ∧ 𝑈𝐽𝐴𝑈) → ∃𝑥𝐽 (𝐴𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3058  wrex 3067  wss 3947  cfv 6548  Topctop 22808  clsccl 22935  Regcreg 23226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-reg 23233
This theorem is referenced by:  regsep2  23293  regr1lem  23656  kqreglem1  23658  kqreglem2  23659  reghmph  23710  cnextcn  23984
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