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Theorem istopg 20902
Description: Express the predicate "𝐽 is a topology." See istop2g 20903 for another characterization using nonempty finite intersections instead of binary intersections.

Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use 𝑇 to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Assertion
Ref Expression
istopg (𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝐴
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem istopg
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 pweq 4305 . . . . 5 (𝑧 = 𝐽 → 𝒫 𝑧 = 𝒫 𝐽)
2 eleq2 2828 . . . . 5 (𝑧 = 𝐽 → ( 𝑥𝑧 𝑥𝐽))
31, 2raleqbidv 3291 . . . 4 (𝑧 = 𝐽 → (∀𝑥 ∈ 𝒫 𝑧 𝑥𝑧 ↔ ∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽))
4 eleq2 2828 . . . . . 6 (𝑧 = 𝐽 → ((𝑥𝑦) ∈ 𝑧 ↔ (𝑥𝑦) ∈ 𝐽))
54raleqbi1dv 3285 . . . . 5 (𝑧 = 𝐽 → (∀𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
65raleqbi1dv 3285 . . . 4 (𝑧 = 𝐽 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
73, 6anbi12d 749 . . 3 (𝑧 = 𝐽 → ((∀𝑥 ∈ 𝒫 𝑧 𝑥𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) ↔ (∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
8 df-top 20901 . . 3 Top = {𝑧 ∣ (∀𝑥 ∈ 𝒫 𝑧 𝑥𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)}
97, 8elab2g 3493 . 2 (𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
10 df-ral 3055 . . . 4 (∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐽 𝑥𝐽))
11 elpw2g 4976 . . . . . 6 (𝐽𝐴 → (𝑥 ∈ 𝒫 𝐽𝑥𝐽))
1211imbi1d 330 . . . . 5 (𝐽𝐴 → ((𝑥 ∈ 𝒫 𝐽 𝑥𝐽) ↔ (𝑥𝐽 𝑥𝐽)))
1312albidv 1998 . . . 4 (𝐽𝐴 → (∀𝑥(𝑥 ∈ 𝒫 𝐽 𝑥𝐽) ↔ ∀𝑥(𝑥𝐽 𝑥𝐽)))
1410, 13syl5bb 272 . . 3 (𝐽𝐴 → (∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ↔ ∀𝑥(𝑥𝐽 𝑥𝐽)))
1514anbi1d 743 . 2 (𝐽𝐴 → ((∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽) ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
169, 15bitrd 268 1 (𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wcel 2139  wral 3050  cin 3714  wss 3715  𝒫 cpw 4302   cuni 4588  Topctop 20900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342  df-in 3722  df-ss 3729  df-pw 4304  df-top 20901
This theorem is referenced by:  istop2g  20903  uniopn  20904  inopn  20906  tgcl  20975  distop  21001  indistopon  21007  fctop  21010  cctop  21012  ppttop  21013  epttop  21015  mretopd  21098  toponmre  21099  neiptoptop  21137  kgentopon  21543  qtoptop2  21704  filconn  21888  utoptop  22239  neibastop1  32660
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