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Theorem toponsspwpw 22840
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 (TopOnβ€˜π΄) βŠ† 𝒫 𝒫 𝐴

Proof of Theorem toponsspwpw
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabssab 4075 . . . . . . 7 {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦} βŠ† {𝑦 ∣ 𝐴 = βˆͺ 𝑦}
2 eqcom 2732 . . . . . . . 8 (𝐴 = βˆͺ 𝑦 ↔ βˆͺ 𝑦 = 𝐴)
32abbii 2795 . . . . . . 7 {𝑦 ∣ 𝐴 = βˆͺ 𝑦} = {𝑦 ∣ βˆͺ 𝑦 = 𝐴}
41, 3sseqtri 4009 . . . . . 6 {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦} βŠ† {𝑦 ∣ βˆͺ 𝑦 = 𝐴}
5 pwpwssunieq 5102 . . . . . 6 {𝑦 ∣ βˆͺ 𝑦 = 𝐴} βŠ† 𝒫 𝒫 𝐴
64, 5sstri 3982 . . . . 5 {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦} βŠ† 𝒫 𝒫 𝐴
7 pwexg 5372 . . . . . 6 (𝐴 ∈ V β†’ 𝒫 𝐴 ∈ V)
87pwexd 5373 . . . . 5 (𝐴 ∈ V β†’ 𝒫 𝒫 𝐴 ∈ V)
9 ssexg 5318 . . . . 5 (({𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦} βŠ† 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) β†’ {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦} ∈ V)
106, 8, 9sylancr 585 . . . 4 (𝐴 ∈ V β†’ {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦} ∈ V)
11 eqeq1 2729 . . . . . 6 (π‘₯ = 𝐴 β†’ (π‘₯ = βˆͺ 𝑦 ↔ 𝐴 = βˆͺ 𝑦))
1211rabbidv 3427 . . . . 5 (π‘₯ = 𝐴 β†’ {𝑦 ∈ Top ∣ π‘₯ = βˆͺ 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦})
13 df-topon 22829 . . . . 5 TopOn = (π‘₯ ∈ V ↦ {𝑦 ∈ Top ∣ π‘₯ = βˆͺ 𝑦})
1412, 13fvmptg 6997 . . . 4 ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦} ∈ V) β†’ (TopOnβ€˜π΄) = {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦})
1510, 14mpdan 685 . . 3 (𝐴 ∈ V β†’ (TopOnβ€˜π΄) = {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦})
1615, 6eqsstrdi 4027 . 2 (𝐴 ∈ V β†’ (TopOnβ€˜π΄) βŠ† 𝒫 𝒫 𝐴)
17 fvprc 6883 . . 3 (Β¬ 𝐴 ∈ V β†’ (TopOnβ€˜π΄) = βˆ…)
18 0ss 4392 . . 3 βˆ… βŠ† 𝒫 𝒫 𝐴
1917, 18eqsstrdi 4027 . 2 (Β¬ 𝐴 ∈ V β†’ (TopOnβ€˜π΄) βŠ† 𝒫 𝒫 𝐴)
2016, 19pm2.61i 182 1 (TopOnβ€˜π΄) βŠ† 𝒫 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1533   ∈ wcel 2098  {cab 2702  {crab 3419  Vcvv 3463   βŠ† wss 3940  βˆ…c0 4318  π’« cpw 4598  βˆͺ cuni 4903  β€˜cfv 6542  Topctop 22811  TopOnctopon 22828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-topon 22829
This theorem is referenced by:  toponmre  23013
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