Theorem isreg 23226

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 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))

Distinct variable group:   𝑥,𝑦,𝑧,𝐽

Proof of Theorem isreg

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6861	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽))
2	1	fveq1d 6863	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ((cls‘𝑗)‘𝑧) = ((cls‘𝐽)‘𝑧))
3	2	sseq1d 3981	. . . . . 6 ⊢ (𝑗 = 𝐽 → (((cls‘𝑗)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
4	3	anbi2d 630	. . . . 5 ⊢ (𝑗 = 𝐽 → ((𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
5	4	rexeqbi1dv 3314	. . . 4 ⊢ (𝑗 = 𝐽 → (∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
6	5	ralbidv 3157	. . 3 ⊢ (𝑗 = 𝐽 → (∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
7	6	raleqbi1dv 3313	. 2 ⊢ (𝑗 = 𝐽 → (∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
8		df-reg 23210	. 2 ⊢ Reg = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
9	7, 8	elrab2 3665	1 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917  ‘cfv 6514  Topctop 22787  clsccl 22912  Regcreg 23203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-reg 23210

This theorem is referenced by:  regtop  23227  regsep  23228  isreg2  23271  kqreglem1  23635  kqreglem2  23636  nrmr0reg  23643  reghmph  23687  utopreg  24147