Theorem toponsspwpwg 12178

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 2692	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		rabssab 3179	. . . . . 6 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ 𝐴 = ∪ 𝑦}
3		eqcom 2139	. . . . . . 7 ⊢ (𝐴 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝐴)
4	3	abbii 2253	. . . . . 6 ⊢ {𝑦 ∣ 𝐴 = ∪ 𝑦} = {𝑦 ∣ ∪ 𝑦 = 𝐴}
5	2, 4	sseqtri 3126	. . . . 5 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝐴}
6		pwpwssunieq 3896	. . . . 5 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴
7	5, 6	sstri 3101	. . . 4 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴
8		pwexg 4099	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
9	8	pwexd 4100	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝒫 𝐴 ∈ V)
10		ssexg 4062	. . . 4 ⊢ (({𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) → {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V)
11	7, 9, 10	sylancr 410	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V)
12		eqeq1 2144	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = ∪ 𝑦 ↔ 𝐴 = ∪ 𝑦))
13	12	rabbidv 2670	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦})
14		df-topon 12167	. . . 4 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦})
15	13, 14	fvmptg 5490	. . 3 ⊢ ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V) → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦})
16	1, 11, 15	syl2anc 408	. 2 ⊢ (𝐴 ∈ 𝑉 → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦})
17	16, 7	eqsstrdi 3144	1 ⊢ (𝐴 ∈ 𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)

Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  {cab 2123  {crab 2418  Vcvv 2681   ⊆ wss 3066  𝒫 cpw 3505  ∪ cuni 3731  ‘cfv 5118  Topctop 12153  TopOnctopon 12166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-topon 12167

This theorem is referenced by: (None)