Theorem toponsspwpwg 14745

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 2814	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		rabssab 3315	. . . . . 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 3261	. . . . 5 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝐴}
6		pwpwssunieq 4059	. . . . 5 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴
7	5, 6	sstri 3236	. . . 4 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴
8		pwexg 4270	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
9	8	pwexd 4271	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝒫 𝐴 ∈ V)
10		ssexg 4228	. . . 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 2791	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦})
14		df-topon 14734	. . . 4 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦})
15	13, 14	fvmptg 5722	. . 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 3279	1 ⊢ (𝐴 ∈ 𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  {cab 2217  {crab 2514  Vcvv 2802   ⊆ wss 3200  𝒫 cpw 3652  ∪ cuni 3893  ‘cfv 5326  Topctop 14720  TopOnctopon 14733

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-topon 14734

This theorem is referenced by: (None)