Theorem tposres0 48858

Description: The transposition of a set restricted to the empty set is the set restricted to the empty set. See also ressn 6246 and dftpos6 48856 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.

Step	Hyp	Ref	Expression
1		relres 5965	. 2 ⊢ Rel (tpos 𝐹 ↾ {∅})
2		relres 5965	. 2 ⊢ Rel (𝐹 ↾ {∅})
3		velsn 4601	. . . . 5 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4		brtpos0 8189	. . . . . . 7 ⊢ (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
5	4	elv 3449	. . . . . 6 ⊢ (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
6		breq1 5105	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
7		breq1 5105	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥𝐹𝑦 ↔ ∅𝐹𝑦))
8	6, 7	bibi12d 345	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥tpos 𝐹𝑦 ↔ 𝑥𝐹𝑦) ↔ (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)))
9	5, 8	mpbiri 258	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥tpos 𝐹𝑦 ↔ 𝑥𝐹𝑦))
10	3, 9	sylbi 217	. . . 4 ⊢ (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ 𝑥𝐹𝑦))
11	10	pm5.32i 574	. . 3 ⊢ ((𝑥 ∈ {∅} ∧ 𝑥tpos 𝐹𝑦) ↔ (𝑥 ∈ {∅} ∧ 𝑥𝐹𝑦))
12		vex 3448	. . . 4 ⊢ 𝑦 ∈ V
13	12	brresi 5948	. . 3 ⊢ (𝑥(tpos 𝐹 ↾ {∅})𝑦 ↔ (𝑥 ∈ {∅} ∧ 𝑥tpos 𝐹𝑦))
14	12	brresi 5948	. . 3 ⊢ (𝑥(𝐹 ↾ {∅})𝑦 ↔ (𝑥 ∈ {∅} ∧ 𝑥𝐹𝑦))
15	11, 13, 14	3bitr4i 303	. 2 ⊢ (𝑥(tpos 𝐹 ↾ {∅})𝑦 ↔ 𝑥(𝐹 ↾ {∅})𝑦)
16	1, 2, 15	eqbrriv 5745	1 ⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444  ∅c0 4292  {csn 4585   class class class wbr 5102   ↾ cres 5633  tpos ctpos 8181

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 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-fv 6507  df-tpos 8182

This theorem is referenced by:  tposresg  48859