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Theorem toponsspwpw 21534
 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.
StepHypRef Expression
1 rabssab 4011 . . . . . . 7 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ {𝑦𝐴 = 𝑦}
2 eqcom 2805 . . . . . . . 8 (𝐴 = 𝑦 𝑦 = 𝐴)
32abbii 2863 . . . . . . 7 {𝑦𝐴 = 𝑦} = {𝑦 𝑦 = 𝐴}
41, 3sseqtri 3951 . . . . . 6 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ {𝑦 𝑦 = 𝐴}
5 pwpwssunieq 4989 . . . . . 6 {𝑦 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴
64, 5sstri 3924 . . . . 5 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ 𝒫 𝒫 𝐴
7 pwexg 5244 . . . . . 6 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
87pwexd 5245 . . . . 5 (𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
9 ssexg 5191 . . . . 5 (({𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) → {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V)
106, 8, 9sylancr 590 . . . 4 (𝐴 ∈ V → {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V)
11 eqeq1 2802 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
1211rabbidv 3427 . . . . 5 (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
13 df-topon 21523 . . . . 5 TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = 𝑦})
1412, 13fvmptg 6743 . . . 4 ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V) → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
1510, 14mpdan 686 . . 3 (𝐴 ∈ V → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
1615, 6eqsstrdi 3969 . 2 (𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
17 fvprc 6638 . . 3 𝐴 ∈ V → (TopOn‘𝐴) = ∅)
18 0ss 4304 . . 3 ∅ ⊆ 𝒫 𝒫 𝐴
1917, 18eqsstrdi 3969 . 2 𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
2016, 19pm2.61i 185 1 (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  {cab 2776  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  ∪ cuni 4800  ‘cfv 6324  Topctop 21505  TopOnctopon 21522 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-topon 21523 This theorem is referenced by:  toponmre  21705
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