Theorem istopg 13077

Description: Express the predicate "𝐽 is a topology". See istopfin 13078 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 3575	. . . . 5 ⊢ (𝑧 = 𝐽 → 𝒫 𝑧 = 𝒫 𝐽)
2		eleq2 2239	. . . . 5 ⊢ (𝑧 = 𝐽 → (∪ 𝑥 ∈ 𝑧 ↔ ∪ 𝑥 ∈ 𝐽))
3	1, 2	raleqbidv 2682	. . . 4 ⊢ (𝑧 = 𝐽 → (∀𝑥 ∈ 𝒫 𝑧∪ 𝑥 ∈ 𝑧 ↔ ∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽))
4		eleq2 2239	. . . . . 6 ⊢ (𝑧 = 𝐽 → ((𝑥 ∩ 𝑦) ∈ 𝑧 ↔ (𝑥 ∩ 𝑦) ∈ 𝐽))
5	4	raleqbi1dv 2678	. . . . 5 ⊢ (𝑧 = 𝐽 → (∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))
6	5	raleqbi1dv 2678	. . . 4 ⊢ (𝑧 = 𝐽 → (∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧 ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))
7	3, 6	anbi12d 473	. . 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 13076	. . 3 ⊢ Top = {𝑧 ∣ (∀𝑥 ∈ 𝒫 𝑧∪ 𝑥 ∈ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)}
9	7, 8	elab2g 2882	. 2 ⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)))
10		df-ral 2458	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽))
11		elpw2g 4151	. . . . . 6 ⊢ (𝐽 ∈ 𝐴 → (𝑥 ∈ 𝒫 𝐽 ↔ 𝑥 ⊆ 𝐽))
12	11	imbi1d 231	. . . . 5 ⊢ (𝐽 ∈ 𝐴 → ((𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽) ↔ (𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽)))
13	12	albidv 1822	. . . 4 ⊢ (𝐽 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽) ↔ ∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽)))
14	10, 13	bitrid 192	. . 3 ⊢ (𝐽 ∈ 𝐴 → (∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ↔ ∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽)))
15	14	anbi1d 465	. 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 188	1 ⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351   = wceq 1353   ∈ wcel 2146  ∀wral 2453   ∩ cin 3126   ⊆ wss 3127  𝒫 cpw 3572  ∪ cuni 3805  Topctop 13075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-sep 4116

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-v 2737  df-in 3133  df-ss 3140  df-pw 3574  df-top 13076

This theorem is referenced by:  istopfin  13078  uniopn  13079  inopn  13081  tgcl  13144  distop  13165  epttop  13170