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Theorem toponsspwpwg 13413
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 2748 . . 3 (𝐴𝑉𝐴 ∈ V)
2 rabssab 3243 . . . . . 6 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ {𝑦𝐴 = 𝑦}
3 eqcom 2179 . . . . . . 7 (𝐴 = 𝑦 𝑦 = 𝐴)
43abbii 2293 . . . . . 6 {𝑦𝐴 = 𝑦} = {𝑦 𝑦 = 𝐴}
52, 4sseqtri 3189 . . . . 5 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ {𝑦 𝑦 = 𝐴}
6 pwpwssunieq 3975 . . . . 5 {𝑦 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴
75, 6sstri 3164 . . . 4 {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ 𝒫 𝒫 𝐴
8 pwexg 4180 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
98pwexd 4181 . . . 4 (𝐴𝑉 → 𝒫 𝒫 𝐴 ∈ V)
10 ssexg 4142 . . . 4 (({𝑦 ∈ Top ∣ 𝐴 = 𝑦} ⊆ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) → {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V)
117, 9, 10sylancr 414 . . 3 (𝐴𝑉 → {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V)
12 eqeq1 2184 . . . . 5 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
1312rabbidv 2726 . . . 4 (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
14 df-topon 13402 . . . 4 TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = 𝑦})
1513, 14fvmptg 5592 . . 3 ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = 𝑦} ∈ V) → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
161, 11, 15syl2anc 411 . 2 (𝐴𝑉 → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = 𝑦})
1716, 7eqsstrdi 3207 1 (𝐴𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  {cab 2163  {crab 2459  Vcvv 2737  wss 3129  𝒫 cpw 3575   cuni 3809  cfv 5216  Topctop 13388  TopOnctopon 13401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-topon 13402
This theorem is referenced by: (None)
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