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Theorem toponsspwpw 22943
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 4094 . . . . . . 7 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ {𝑦𝐴 = 𝑦}
2 eqcom 2741 . . . . . . . 8 (𝐴 = 𝑦 𝑦 = 𝐴)
32abbii 2806 . . . . . . 7 {𝑦𝐴 = 𝑦} = {𝑦 𝑦 = 𝐴}
41, 3sseqtri 4031 . . . . . 6 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ {𝑦 𝑦 = 𝐴}
5 pwpwssunieq 5108 . . . . . 6 {𝑦 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴
64, 5sstri 4004 . . . . 5 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ 𝒫 𝒫 𝐴
7 pwexg 5383 . . . . . 6 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
87pwexd 5384 . . . . 5 (𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
9 ssexg 5328 . . . . 5 (({𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) → {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V)
106, 8, 9sylancr 587 . . . 4 (𝐴 ∈ V → {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V)
11 eqeq1 2738 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
1211rabbidv 3440 . . . . 5 (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
13 df-topon 22932 . . . . 5 TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = 𝑦})
1412, 13fvmptg 7013 . . . 4 ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V) → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
1510, 14mpdan 687 . . 3 (𝐴 ∈ V → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
1615, 6eqsstrdi 4049 . 2 (𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
17 fvprc 6898 . . 3 𝐴 ∈ V → (TopOn‘𝐴) = ∅)
18 0ss 4405 . . 3 ∅ ⊆ 𝒫 𝒫 𝐴
1917, 18eqsstrdi 4049 . 2 𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
2016, 19pm2.61i 182 1 (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1536  wcel 2105  {cab 2711  {crab 3432  Vcvv 3477  wss 3962  c0 4338  𝒫 cpw 4604   cuni 4911  cfv 6562  Topctop 22914  TopOnctopon 22931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-topon 22932
This theorem is referenced by:  toponmre  23116
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