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Theorem istopg 22397
Description: Express the predicate "𝐽 is a topology". See istop2g 22398 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 4617 . . . . 5 (𝑧 = 𝐽 → 𝒫 𝑧 = 𝒫 𝐽)
2 eleq2 2823 . . . . 5 (𝑧 = 𝐽 → ( 𝑥𝑧 𝑥𝐽))
31, 2raleqbidv 3343 . . . 4 (𝑧 = 𝐽 → (∀𝑥 ∈ 𝒫 𝑧 𝑥𝑧 ↔ ∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽))
4 eleq2 2823 . . . . . 6 (𝑧 = 𝐽 → ((𝑥𝑦) ∈ 𝑧 ↔ (𝑥𝑦) ∈ 𝐽))
54raleqbi1dv 3334 . . . . 5 (𝑧 = 𝐽 → (∀𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
65raleqbi1dv 3334 . . . 4 (𝑧 = 𝐽 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
73, 6anbi12d 632 . . 3 (𝑧 = 𝐽 → ((∀𝑥 ∈ 𝒫 𝑧 𝑥𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) ↔ (∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
8 df-top 22396 . . 3 Top = {𝑧 ∣ (∀𝑥 ∈ 𝒫 𝑧 𝑥𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)}
97, 8elab2g 3671 . 2 (𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
10 df-ral 3063 . . . 4 (∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐽 𝑥𝐽))
11 elpw2g 5345 . . . . . 6 (𝐽𝐴 → (𝑥 ∈ 𝒫 𝐽𝑥𝐽))
1211imbi1d 342 . . . . 5 (𝐽𝐴 → ((𝑥 ∈ 𝒫 𝐽 𝑥𝐽) ↔ (𝑥𝐽 𝑥𝐽)))
1312albidv 1924 . . . 4 (𝐽𝐴 → (∀𝑥(𝑥 ∈ 𝒫 𝐽 𝑥𝐽) ↔ ∀𝑥(𝑥𝐽 𝑥𝐽)))
1410, 13bitrid 283 . . 3 (𝐽𝐴 → (∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ↔ ∀𝑥(𝑥𝐽 𝑥𝐽)))
1514anbi1d 631 . 2 (𝐽𝐴 → ((∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽) ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
169, 15bitrd 279 1 (𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wcel 2107  wral 3062  cin 3948  wss 3949  𝒫 cpw 4603   cuni 4909  Topctop 22395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-pw 4605  df-top 22396
This theorem is referenced by:  istop2g  22398  uniopn  22399  inopn  22401  tgcl  22472  distop  22498  indistopon  22504  fctop  22507  cctop  22509  ppttop  22510  epttop  22512  mretopd  22596  toponmre  22597  neiptoptop  22635  kgentopon  23042  qtoptop2  23203  filconn  23387  utoptop  23739  neibastop1  35292
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