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Theorem isreg 20946
 Description: The predicate "is a regular space." In a regular space, any open neighborhood has a closed subneighborhood. Note that some authors require the space to be Hausdorff (which would make it the same as T3), but we reserve the phrase "regular Hausdorff" for that as many topologists do. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
isreg (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
Distinct variable group:   𝑥,𝑦,𝑧,𝐽

Proof of Theorem isreg
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . . . . . . 8 (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽))
21fveq1d 6105 . . . . . . 7 (𝑗 = 𝐽 → ((cls‘𝑗)‘𝑧) = ((cls‘𝐽)‘𝑧))
32sseq1d 3595 . . . . . 6 (𝑗 = 𝐽 → (((cls‘𝑗)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
43anbi2d 736 . . . . 5 (𝑗 = 𝐽 → ((𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
54rexeqbi1dv 3124 . . . 4 (𝑗 = 𝐽 → (∃𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∃𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
65ralbidv 2969 . . 3 (𝑗 = 𝐽 → (∀𝑦𝑥𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∀𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
76raleqbi1dv 3123 . 2 (𝑗 = 𝐽 → (∀𝑥𝑗𝑦𝑥𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∀𝑥𝐽𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
8 df-reg 20930 . 2 Reg = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
97, 8elrab2 3333 1 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ‘cfv 5804  Topctop 20517  clsccl 20632  Regcreg 20923 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-reg 20930 This theorem is referenced by:  regtop  20947  regsep  20948  isreg2  20991  kqreglem1  21354  kqreglem2  21355  nrmr0reg  21362  reghmph  21406  utopreg  21866
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