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Theorem regsep 21078
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 21076 . . . 4 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑦𝐽𝑧𝑦𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦)))
2 sseq2 3612 . . . . . . . 8 (𝑦 = 𝑈 → (((cls‘𝐽)‘𝑥) ⊆ 𝑦 ↔ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))
32anbi2d 739 . . . . . . 7 (𝑦 = 𝑈 → ((𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
43rexbidv 3047 . . . . . 6 (𝑦 = 𝑈 → (∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
54raleqbi1dv 3139 . . . . 5 (𝑦 = 𝑈 → (∀𝑧𝑦𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∀𝑧𝑈𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
65rspccv 3296 . . . 4 (∀𝑦𝐽𝑧𝑦𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) → (𝑈𝐽 → ∀𝑧𝑈𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
71, 6simplbiim 658 . . 3 (𝐽 ∈ Reg → (𝑈𝐽 → ∀𝑧𝑈𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
8 eleq1 2686 . . . . . 6 (𝑧 = 𝐴 → (𝑧𝑥𝐴𝑥))
98anbi1d 740 . . . . 5 (𝑧 = 𝐴 → ((𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈) ↔ (𝐴𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
109rexbidv 3047 . . . 4 (𝑧 = 𝐴 → (∃𝑥𝐽 (𝑧𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈) ↔ ∃𝑥𝐽 (𝐴𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
1110rspccv 3296 . . 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 1254 1 ((𝐽 ∈ Reg ∧ 𝑈𝐽𝐴𝑈) → ∃𝑥𝐽 (𝐴𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  wrex 2909  wss 3560  cfv 5857  Topctop 20638  clsccl 20762  Regcreg 21053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-reg 21060
This theorem is referenced by:  regsep2  21120  regr1lem  21482  kqreglem1  21484  kqreglem2  21485  reghmph  21536  cnextcn  21811
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