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Theorem toponsspwpw 22287
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 4048 . . . . . . 7 {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦} βŠ† {𝑦 ∣ 𝐴 = βˆͺ 𝑦}
2 eqcom 2744 . . . . . . . 8 (𝐴 = βˆͺ 𝑦 ↔ βˆͺ 𝑦 = 𝐴)
32abbii 2807 . . . . . . 7 {𝑦 ∣ 𝐴 = βˆͺ 𝑦} = {𝑦 ∣ βˆͺ 𝑦 = 𝐴}
41, 3sseqtri 3985 . . . . . 6 {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦} βŠ† {𝑦 ∣ βˆͺ 𝑦 = 𝐴}
5 pwpwssunieq 5069 . . . . . 6 {𝑦 ∣ βˆͺ 𝑦 = 𝐴} βŠ† 𝒫 𝒫 𝐴
64, 5sstri 3958 . . . . 5 {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦} βŠ† 𝒫 𝒫 𝐴
7 pwexg 5338 . . . . . 6 (𝐴 ∈ V β†’ 𝒫 𝐴 ∈ V)
87pwexd 5339 . . . . 5 (𝐴 ∈ V β†’ 𝒫 𝒫 𝐴 ∈ V)
9 ssexg 5285 . . . . 5 (({𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦} βŠ† 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) β†’ {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦} ∈ V)
106, 8, 9sylancr 588 . . . 4 (𝐴 ∈ V β†’ {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦} ∈ V)
11 eqeq1 2741 . . . . . 6 (π‘₯ = 𝐴 β†’ (π‘₯ = βˆͺ 𝑦 ↔ 𝐴 = βˆͺ 𝑦))
1211rabbidv 3418 . . . . 5 (π‘₯ = 𝐴 β†’ {𝑦 ∈ Top ∣ π‘₯ = βˆͺ 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦})
13 df-topon 22276 . . . . 5 TopOn = (π‘₯ ∈ V ↦ {𝑦 ∈ Top ∣ π‘₯ = βˆͺ 𝑦})
1412, 13fvmptg 6951 . . . 4 ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦} ∈ V) β†’ (TopOnβ€˜π΄) = {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦})
1510, 14mpdan 686 . . 3 (𝐴 ∈ V β†’ (TopOnβ€˜π΄) = {𝑦 ∈ Top ∣ 𝐴 = βˆͺ 𝑦})
1615, 6eqsstrdi 4003 . 2 (𝐴 ∈ V β†’ (TopOnβ€˜π΄) βŠ† 𝒫 𝒫 𝐴)
17 fvprc 6839 . . 3 (Β¬ 𝐴 ∈ V β†’ (TopOnβ€˜π΄) = βˆ…)
18 0ss 4361 . . 3 βˆ… βŠ† 𝒫 𝒫 𝐴
1917, 18eqsstrdi 4003 . 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 1542   ∈ wcel 2107  {cab 2714  {crab 3410  Vcvv 3448   βŠ† wss 3915  βˆ…c0 4287  π’« cpw 4565  βˆͺ cuni 4870  β€˜cfv 6501  Topctop 22258  TopOnctopon 22275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-topon 22276
This theorem is referenced by:  toponmre  22460
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