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Theorem toponsspwpw 22912
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 4079 . . . . . . 7 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ {𝑦𝐴 = 𝑦}
2 eqcom 2733 . . . . . . . 8 (𝐴 = 𝑦 𝑦 = 𝐴)
32abbii 2796 . . . . . . 7 {𝑦𝐴 = 𝑦} = {𝑦 𝑦 = 𝐴}
41, 3sseqtri 4015 . . . . . 6 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ {𝑦 𝑦 = 𝐴}
5 pwpwssunieq 5104 . . . . . 6 {𝑦 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴
64, 5sstri 3988 . . . . 5 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ 𝒫 𝒫 𝐴
7 pwexg 5374 . . . . . 6 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
87pwexd 5375 . . . . 5 (𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
9 ssexg 5320 . . . . 5 (({𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) → {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V)
106, 8, 9sylancr 585 . . . 4 (𝐴 ∈ V → {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V)
11 eqeq1 2730 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
1211rabbidv 3427 . . . . 5 (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
13 df-topon 22901 . . . . 5 TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = 𝑦})
1412, 13fvmptg 6999 . . . 4 ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V) → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
1510, 14mpdan 685 . . 3 (𝐴 ∈ V → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
1615, 6eqsstrdi 4033 . 2 (𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
17 fvprc 6885 . . 3 𝐴 ∈ V → (TopOn‘𝐴) = ∅)
18 0ss 4394 . . 3 ∅ ⊆ 𝒫 𝒫 𝐴
1917, 18eqsstrdi 4033 . 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 1534  wcel 2099  {cab 2703  {crab 3419  Vcvv 3462  wss 3946  c0 4322  𝒫 cpw 4597   cuni 4905  cfv 6546  Topctop 22883  TopOnctopon 22900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-iota 6498  df-fun 6548  df-fv 6554  df-topon 22901
This theorem is referenced by:  toponmre  23085
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