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Theorem istopg 20525
 Description: Express the predicate "𝐽 is a topology." Note: In the literature, a topology is often represented by a script 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 4111 . . . . 5 (𝑧 = 𝐽 → 𝒫 𝑧 = 𝒫 𝐽)
2 eleq2 2677 . . . . 5 (𝑧 = 𝐽 → ( 𝑥𝑧 𝑥𝐽))
31, 2raleqbidv 3129 . . . 4 (𝑧 = 𝐽 → (∀𝑥 ∈ 𝒫 𝑧 𝑥𝑧 ↔ ∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽))
4 eleq2 2677 . . . . . 6 (𝑧 = 𝐽 → ((𝑥𝑦) ∈ 𝑧 ↔ (𝑥𝑦) ∈ 𝐽))
54raleqbi1dv 3123 . . . . 5 (𝑧 = 𝐽 → (∀𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
65raleqbi1dv 3123 . . . 4 (𝑧 = 𝐽 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
73, 6anbi12d 743 . . 3 (𝑧 = 𝐽 → ((∀𝑥 ∈ 𝒫 𝑧 𝑥𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) ↔ (∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
8 df-top 20521 . . 3 Top = {𝑧 ∣ (∀𝑥 ∈ 𝒫 𝑧 𝑥𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)}
97, 8elab2g 3322 . 2 (𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
10 df-ral 2901 . . . 4 (∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐽 𝑥𝐽))
11 elpw2g 4754 . . . . . 6 (𝐽𝐴 → (𝑥 ∈ 𝒫 𝐽𝑥𝐽))
1211imbi1d 330 . . . . 5 (𝐽𝐴 → ((𝑥 ∈ 𝒫 𝐽 𝑥𝐽) ↔ (𝑥𝐽 𝑥𝐽)))
1312albidv 1836 . . . 4 (𝐽𝐴 → (∀𝑥(𝑥 ∈ 𝒫 𝐽 𝑥𝐽) ↔ ∀𝑥(𝑥𝐽 𝑥𝐽)))
1410, 13syl5bb 271 . . 3 (𝐽𝐴 → (∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ↔ ∀𝑥(𝑥𝐽 𝑥𝐽)))
1514anbi1d 737 . 2 (𝐽𝐴 → ((∀𝑥 ∈ 𝒫 𝐽 𝑥𝐽 ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽) ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
169, 15bitrd 267 1 (𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  Topctop 20517 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  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-top 20521 This theorem is referenced by:  istop2g  20526  uniopn  20527  inopn  20529  tgcl  20584  distop  20610  indistopon  20615  fctop  20618  cctop  20620  ppttop  20621  epttop  20623  mretopd  20706  toponmre  20707  neiptoptop  20745  kgentopon  21151  qtoptop2  21312  filcon  21497  utoptop  21848  neibastop1  31524
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