Theorem toponsspwpwg 14816

Description: The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)

Assertion

Ref	Expression
toponsspwpwg	⊢ (𝐴 ∈ 𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)

Proof of Theorem toponsspwpwg

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

Step	Hyp	Ref	Expression
1		elex 2815	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		rabssab 3317	. . . . . 6 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ 𝐴 = ∪ 𝑦}
3		eqcom 2233	. . . . . . 7 ⊢ (𝐴 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝐴)
4	3	abbii 2347	. . . . . 6 ⊢ {𝑦 ∣ 𝐴 = ∪ 𝑦} = {𝑦 ∣ ∪ 𝑦 = 𝐴}
5	2, 4	sseqtri 3262	. . . . 5 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝐴}
6		pwpwssunieq 4064	. . . . 5 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴
7	5, 6	sstri 3237	. . . 4 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴
8		pwexg 4276	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
9	8	pwexd 4277	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝒫 𝐴 ∈ V)
10		ssexg 4233	. . . 4 ⊢ (({𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) → {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V)
11	7, 9, 10	sylancr 414	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V)
12		eqeq1 2238	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = ∪ 𝑦 ↔ 𝐴 = ∪ 𝑦))
13	12	rabbidv 2792	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦})
14		df-topon 14805	. . . 4 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦})
15	13, 14	fvmptg 5731	. . 3 ⊢ ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V) → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦})
16	1, 11, 15	syl2anc 411	. 2 ⊢ (𝐴 ∈ 𝑉 → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦})
17	16, 7	eqsstrdi 3280	1 ⊢ (𝐴 ∈ 𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2202  {cab 2217  {crab 2515  Vcvv 2803   ⊆ wss 3201  𝒫 cpw 3656  ∪ cuni 3898  ‘cfv 5333  Topctop 14791  TopOnctopon 14804

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-topon 14805

This theorem is referenced by: (None)