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Theorem bj-toponss 32694
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 3668 . . . . . . 7 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ {𝑦𝐴 = 𝑦}
2 eqcom 2628 . . . . . . . 8 (𝐴 = 𝑦 𝑦 = 𝐴)
32abbii 2736 . . . . . . 7 {𝑦𝐴 = 𝑦} = {𝑦 𝑦 = 𝐴}
41, 3sseqtri 3616 . . . . . 6 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ {𝑦 𝑦 = 𝐴}
5 bj-sspwpweq 32693 . . . . . 6 {𝑦 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴
64, 5sstri 3592 . . . . 5 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ 𝒫 𝒫 𝐴
7 pwexg 4810 . . . . . 6 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
8 pwexg 4810 . . . . . 6 (𝒫 𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
97, 8syl 17 . . . . 5 (𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
10 ssexg 4764 . . . . 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 2625 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
1312rabbidv 3177 . . . . 5 (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
14 df-topon 20623 . . . . 5 TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = 𝑦})
1513, 14fvmptg 6237 . . . 4 ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V) → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
1611, 15mpdan 701 . . 3 (𝐴 ∈ V → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
1716, 6syl6eqss 3634 . 2 (𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
18 fvprc 6142 . . 3 𝐴 ∈ V → (TopOn‘𝐴) = ∅)
19 0ss 3944 . . 3 ∅ ⊆ 𝒫 𝒫 𝐴
2018, 19syl6eqss 3634 . 2 𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
2117, 20pm2.61i 176 1 (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  {cab 2607  {crab 2911  Vcvv 3186  wss 3555  c0 3891  𝒫 cpw 4130   cuni 4402  cfv 5847  Topctop 20617  TopOnctopon 20618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-topon 20623
This theorem is referenced by: (None)
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