Theorem istopg 22044

Description: Express the predicate "𝐽 is a topology". See istop2g 22045 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	⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 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.

Step	Hyp	Ref	Expression
1		pweq 4549	. . . . 5 ⊢ (𝑧 = 𝐽 → 𝒫 𝑧 = 𝒫 𝐽)
2		eleq2 2827	. . . . 5 ⊢ (𝑧 = 𝐽 → (∪ 𝑥 ∈ 𝑧 ↔ ∪ 𝑥 ∈ 𝐽))
3	1, 2	raleqbidv 3336	. . . 4 ⊢ (𝑧 = 𝐽 → (∀𝑥 ∈ 𝒫 𝑧∪ 𝑥 ∈ 𝑧 ↔ ∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽))
4		eleq2 2827	. . . . . 6 ⊢ (𝑧 = 𝐽 → ((𝑥 ∩ 𝑦) ∈ 𝑧 ↔ (𝑥 ∩ 𝑦) ∈ 𝐽))
5	4	raleqbi1dv 3340	. . . . 5 ⊢ (𝑧 = 𝐽 → (∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))
6	5	raleqbi1dv 3340	. . . 4 ⊢ (𝑧 = 𝐽 → (∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧 ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))
7	3, 6	anbi12d 631	. . 3 ⊢ (𝑧 = 𝐽 → ((∀𝑥 ∈ 𝒫 𝑧∪ 𝑥 ∈ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧) ↔ (∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)))
8		df-top 22043	. . 3 ⊢ Top = {𝑧 ∣ (∀𝑥 ∈ 𝒫 𝑧∪ 𝑥 ∈ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)}
9	7, 8	elab2g 3611	. 2 ⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)))
10		df-ral 3069	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽))
11		elpw2g 5268	. . . . . 6 ⊢ (𝐽 ∈ 𝐴 → (𝑥 ∈ 𝒫 𝐽 ↔ 𝑥 ⊆ 𝐽))
12	11	imbi1d 342	. . . . 5 ⊢ (𝐽 ∈ 𝐴 → ((𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽) ↔ (𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽)))
13	12	albidv 1923	. . . 4 ⊢ (𝐽 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽) ↔ ∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽)))
14	10, 13	bitrid 282	. . 3 ⊢ (𝐽 ∈ 𝐴 → (∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ↔ ∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽)))
15	14	anbi1d 630	. 2 ⊢ (𝐽 ∈ 𝐴 → ((∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽) ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)))
16	9, 15	bitrd 278	1 ⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839  Topctop 22042

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-pw 4535  df-top 22043

This theorem is referenced by:  istop2g  22045  uniopn  22046  inopn  22048  tgcl  22119  distop  22145  indistopon  22151  fctop  22154  cctop  22156  ppttop  22157  epttop  22159  mretopd  22243  toponmre  22244  neiptoptop  22282  kgentopon  22689  qtoptop2  22850  filconn  23034  utoptop  23386  neibastop1  34548