Theorem toponsspwpw 20926

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 3830	. . . . . . 7 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ 𝐴 = ∪ 𝑦}
2		eqcom 2765	. . . . . . . 8 ⊢ (𝐴 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝐴)
3	2	abbii 2875	. . . . . . 7 ⊢ {𝑦 ∣ 𝐴 = ∪ 𝑦} = {𝑦 ∣ ∪ 𝑦 = 𝐴}
4	1, 3	sseqtri 3776	. . . . . 6 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝐴}
5		pwpwssunieq 4765	. . . . . 6 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴
6	4, 5	sstri 3751	. . . . 5 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴
7		pwexg 4997	. . . . . 6 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
8		pwexg 4997	. . . . . 6 ⊢ (𝒫 𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
9	7, 8	syl 17	. . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
10		ssexg 4954	. . . . 5 ⊢ (({𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) → {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V)
11	6, 9, 10	sylancr 698	. . . 4 ⊢ (𝐴 ∈ V → {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V)
12		eqeq1 2762	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = ∪ 𝑦 ↔ 𝐴 = ∪ 𝑦))
13	12	rabbidv 3327	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦})
14		df-topon 20916	. . . . 5 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦})
15	13, 14	fvmptg 6440	. . . 4 ⊢ ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V) → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦})
16	11, 15	mpdan 705	. . 3 ⊢ (𝐴 ∈ V → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦})
17	16, 6	syl6eqss 3794	. 2 ⊢ (𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
18		fvprc 6344	. . 3 ⊢ (¬ 𝐴 ∈ V → (TopOn‘𝐴) = ∅)
19		0ss 4113	. . 3 ⊢ ∅ ⊆ 𝒫 𝒫 𝐴
20	18, 19	syl6eqss 3794	. 2 ⊢ (¬ 𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
21	17, 20	pm2.61i 176	1 ⊢ (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴

Syntax hints:  ¬ wn 3   = wceq 1630   ∈ wcel 2137  {cab 2744  {crab 3052  Vcvv 3338   ⊆ wss 3713  ∅c0 4056  𝒫 cpw 4300  ∪ cuni 4586  ‘cfv 6047  Topctop 20898  TopOnctopon 20915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-iota 6010  df-fun 6049  df-fv 6055  df-topon 20916

This theorem is referenced by: (None)