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Theorem toponsspwpwg 12228
 Description: The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
Assertion
Ref Expression
toponsspwpwg (𝐴𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)

Proof of Theorem toponsspwpwg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2700 . . 3 (𝐴𝑉𝐴 ∈ V)
2 rabssab 3189 . . . . . 6 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ {𝑦𝐴 = 𝑦}
3 eqcom 2142 . . . . . . 7 (𝐴 = 𝑦 𝑦 = 𝐴)
43abbii 2256 . . . . . 6 {𝑦𝐴 = 𝑦} = {𝑦 𝑦 = 𝐴}
52, 4sseqtri 3136 . . . . 5 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ {𝑦 𝑦 = 𝐴}
6 pwpwssunieq 3909 . . . . 5 {𝑦 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴
75, 6sstri 3111 . . . 4 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ 𝒫 𝒫 𝐴
8 pwexg 4112 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
98pwexd 4113 . . . 4 (𝐴𝑉 → 𝒫 𝒫 𝐴 ∈ V)
10 ssexg 4075 . . . 4 (({𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) → {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V)
117, 9, 10sylancr 411 . . 3 (𝐴𝑉 → {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V)
12 eqeq1 2147 . . . . 5 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
1312rabbidv 2678 . . . 4 (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
14 df-topon 12217 . . . 4 TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = 𝑦})
1513, 14fvmptg 5505 . . 3 ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V) → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
161, 11, 15syl2anc 409 . 2 (𝐴𝑉 → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
1716, 7eqsstrdi 3154 1 (𝐴𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  {cab 2126  {crab 2421  Vcvv 2689   ⊆ wss 3076  𝒫 cpw 3515  ∪ cuni 3744  ‘cfv 5131  Topctop 12203  TopOnctopon 12216 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-topon 12217 This theorem is referenced by: (None)
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