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Theorem tposres0 48855
Description: The transposition of a set restricted to the empty set is the set restricted to the empty set. See also ressn 6260 and dftpos6 48853 for an alternate proof. (Contributed by Zhi Wang, 6-Oct-2025.)
Assertion
Ref Expression
tposres0 (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})

Proof of Theorem tposres0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 5978 . 2 Rel (tpos 𝐹 ↾ {∅})
2 relres 5978 . 2 Rel (𝐹 ↾ {∅})
3 velsn 4607 . . . . 5 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4 brtpos0 8214 . . . . . . 7 (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
54elv 3455 . . . . . 6 (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
6 breq1 5112 . . . . . . 7 (𝑥 = ∅ → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
7 breq1 5112 . . . . . . 7 (𝑥 = ∅ → (𝑥𝐹𝑦 ↔ ∅𝐹𝑦))
86, 7bibi12d 345 . . . . . 6 (𝑥 = ∅ → ((𝑥tpos 𝐹𝑦𝑥𝐹𝑦) ↔ (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)))
95, 8mpbiri 258 . . . . 5 (𝑥 = ∅ → (𝑥tpos 𝐹𝑦𝑥𝐹𝑦))
103, 9sylbi 217 . . . 4 (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦𝑥𝐹𝑦))
1110pm5.32i 574 . . 3 ((𝑥 ∈ {∅} ∧ 𝑥tpos 𝐹𝑦) ↔ (𝑥 ∈ {∅} ∧ 𝑥𝐹𝑦))
12 vex 3454 . . . 4 𝑦 ∈ V
1312brresi 5961 . . 3 (𝑥(tpos 𝐹 ↾ {∅})𝑦 ↔ (𝑥 ∈ {∅} ∧ 𝑥tpos 𝐹𝑦))
1412brresi 5961 . . 3 (𝑥(𝐹 ↾ {∅})𝑦 ↔ (𝑥 ∈ {∅} ∧ 𝑥𝐹𝑦))
1511, 13, 143bitr4i 303 . 2 (𝑥(tpos 𝐹 ↾ {∅})𝑦𝑥(𝐹 ↾ {∅})𝑦)
161, 2, 15eqbrriv 5756 1 (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2109  Vcvv 3450  c0 4298  {csn 4591   class class class wbr 5109  cres 5642  tpos ctpos 8206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-fv 6521  df-tpos 8207
This theorem is referenced by:  tposresg  48856
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