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Theorem istopg 22196
Description: Express the predicate "𝐽 is a topology". See istop2g 22197 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 4572 . . . . 5 (𝑧 = 𝐽 → 𝒫 𝑧 = 𝒫 𝐽)
2 eleq2 2826 . . . . 5 (𝑧 = 𝐽 → ( 𝑥𝑧 𝑥𝐽))
31, 2raleqbidv 3317 . . . 4 (𝑧 = 𝐽 → (∀𝑥 ∈ 𝒫 𝑧 𝑥𝑧 ↔ ∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽))
4 eleq2 2826 . . . . . 6 (𝑧 = 𝐽 → ((𝑥𝑦) ∈ 𝑧 ↔ (𝑥𝑦) ∈ 𝐽))
54raleqbi1dv 3305 . . . . 5 (𝑧 = 𝐽 → (∀𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
65raleqbi1dv 3305 . . . 4 (𝑧 = 𝐽 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
73, 6anbi12d 631 . . 3 (𝑧 = 𝐽 → ((∀𝑥 ∈ 𝒫 𝑧 𝑥𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) ↔ (∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
8 df-top 22195 . . 3 Top = {𝑧 ∣ (∀𝑥 ∈ 𝒫 𝑧 𝑥𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)}
97, 8elab2g 3630 . 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 5299 . . . . . 6 (𝐽𝐴 → (𝑥 ∈ 𝒫 𝐽𝑥𝐽))
1211imbi1d 341 . . . . 5 (𝐽𝐴 → ((𝑥 ∈ 𝒫 𝐽 𝑥𝐽) ↔ (𝑥𝐽 𝑥𝐽)))
1312albidv 1923 . . . 4 (𝐽𝐴 → (∀𝑥(𝑥 ∈ 𝒫 𝐽 𝑥𝐽) ↔ ∀𝑥(𝑥𝐽 𝑥𝐽)))
1410, 13bitrid 282 . . 3 (𝐽𝐴 → (∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ↔ ∀𝑥(𝑥𝐽 𝑥𝐽)))
1514anbi1d 630 . 2 (𝐽𝐴 → ((∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽) ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
169, 15bitrd 278 1 (𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wcel 2106  wral 3062  cin 3907  wss 3908  𝒫 cpw 4558   cuni 4863  Topctop 22194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rab 3406  df-v 3445  df-in 3915  df-ss 3925  df-pw 4560  df-top 22195
This theorem is referenced by:  istop2g  22197  uniopn  22198  inopn  22200  tgcl  22271  distop  22297  indistopon  22303  fctop  22306  cctop  22308  ppttop  22309  epttop  22311  mretopd  22395  toponmre  22396  neiptoptop  22434  kgentopon  22841  qtoptop2  23002  filconn  23186  utoptop  23538  neibastop1  34769
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