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Theorem bj-toponss 32241
 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
bj-toponss (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴

Proof of Theorem bj-toponss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabssab 3652 . . . . . . 7 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ {𝑦𝐴 = 𝑦}
2 eqcom 2617 . . . . . . . 8 (𝐴 = 𝑦 𝑦 = 𝐴)
32abbii 2726 . . . . . . 7 {𝑦𝐴 = 𝑦} = {𝑦 𝑦 = 𝐴}
41, 3sseqtri 3600 . . . . . 6 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ {𝑦 𝑦 = 𝐴}
5 bj-sspwpweq 32240 . . . . . 6 {𝑦 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴
64, 5sstri 3577 . . . . 5 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ 𝒫 𝒫 𝐴
7 pwexg 4776 . . . . . 6 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
8 pwexg 4776 . . . . . 6 (𝒫 𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
97, 8syl 17 . . . . 5 (𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
10 ssexg 4732 . . . . 5 (({𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) → {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V)
116, 9, 10sylancr 694 . . . 4 (𝐴 ∈ V → {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V)
12 eqeq1 2614 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
1312rabbidv 3164 . . . . 5 (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
14 df-topon 20523 . . . . 5 TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = 𝑦})
1513, 14fvmptg 6189 . . . 4 ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V) → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
1611, 15mpdan 699 . . 3 (𝐴 ∈ V → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
1716, 6syl6eqss 3618 . 2 (𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
18 fvprc 6097 . . 3 𝐴 ∈ V → (TopOn‘𝐴) = ∅)
19 0ss 3924 . . 3 ∅ ⊆ 𝒫 𝒫 𝐴
2018, 19syl6eqss 3618 . 2 𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
2117, 20pm2.61i 175 1 (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  TopOnctopon 20518 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-topon 20523 This theorem is referenced by: (None)
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