Theorem toponsspwpw 22807

Description: The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
toponsspwpw	⊢ (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴

Proof of Theorem toponsspwpw

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabssab 4036	. . . . . . 7 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ 𝐴 = ∪ 𝑦}
2		eqcom 2736	. . . . . . . 8 ⊢ (𝐴 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝐴)
3	2	abbii 2796	. . . . . . 7 ⊢ {𝑦 ∣ 𝐴 = ∪ 𝑦} = {𝑦 ∣ ∪ 𝑦 = 𝐴}
4	1, 3	sseqtri 3984	. . . . . 6 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝐴}
5		pwpwssunieq 5053	. . . . . 6 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴
6	4, 5	sstri 3945	. . . . 5 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴
7		pwexg 5317	. . . . . 6 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
8	7	pwexd 5318	. . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
9		ssexg 5262	. . . . 5 ⊢ (({𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) → {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V)
10	6, 8, 9	sylancr 587	. . . 4 ⊢ (𝐴 ∈ V → {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V)
11		eqeq1 2733	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = ∪ 𝑦 ↔ 𝐴 = ∪ 𝑦))
12	11	rabbidv 3402	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦})
13		df-topon 22796	. . . . 5 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦})
14	12, 13	fvmptg 6928	. . . 4 ⊢ ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V) → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦})
15	10, 14	mpdan 687	. . 3 ⊢ (𝐴 ∈ V → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦})
16	15, 6	eqsstrdi 3980	. 2 ⊢ (𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
17		fvprc 6814	. . 3 ⊢ (¬ 𝐴 ∈ V → (TopOn‘𝐴) = ∅)
18		0ss 4351	. . 3 ⊢ ∅ ⊆ 𝒫 𝒫 𝐴
19	17, 18	eqsstrdi 3980	. 2 ⊢ (¬ 𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
20	16, 19	pm2.61i 182	1 ⊢ (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  {cab 2707  {crab 3394  Vcvv 3436   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  ∪ cuni 4858  ‘cfv 6482  Topctop 22778  TopOnctopon 22795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-topon 22796

This theorem is referenced by:  toponmre  22978