Theorem tposres0 49540

Description: The transposition of a set restricted to the empty set is the set restricted to the empty set. See also ressn 6287 and dftpos6 49538 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 6005	. 2 ⊢ Rel (tpos 𝐹 ↾ {∅})
2		relres 6005	. 2 ⊢ Rel (𝐹 ↾ {∅})
3		velsn 4610	. . . . 5 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4		brtpos0 8229	. . . . . . 7 ⊢ (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
5	4	elv 3468	. . . . . 6 ⊢ (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
6		breq1 5116	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
7		breq1 5116	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥𝐹𝑦 ↔ ∅𝐹𝑦))
8	6, 7	bibi12d 348	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥tpos 𝐹𝑦 ↔ 𝑥𝐹𝑦) ↔ (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)))
9	5, 8	mpbiri 261	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥tpos 𝐹𝑦 ↔ 𝑥𝐹𝑦))
10	3, 9	sylbi 220	. . . 4 ⊢ (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ 𝑥𝐹𝑦))
11	10	pm5.32i 584	. . 3 ⊢ ((𝑥 ∈ {∅} ∧ 𝑥tpos 𝐹𝑦) ↔ (𝑥 ∈ {∅} ∧ 𝑥𝐹𝑦))
12		vex 3467	. . . 4 ⊢ 𝑦 ∈ V
13	12	brresi 5988	. . 3 ⊢ (𝑥(tpos 𝐹 ↾ {∅})𝑦 ↔ (𝑥 ∈ {∅} ∧ 𝑥tpos 𝐹𝑦))
14	12	brresi 5988	. . 3 ⊢ (𝑥(𝐹 ↾ {∅})𝑦 ↔ (𝑥 ∈ {∅} ∧ 𝑥𝐹𝑦))
15	11, 13, 14	3bitr4i 306	. 2 ⊢ (𝑥(tpos 𝐹 ↾ {∅})𝑦 ↔ 𝑥(𝐹 ↾ {∅})𝑦)
16	1, 2, 15	eqbrriv 5778	1 ⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ∅c0 4294  {csn 4594   class class class wbr 5113   ↾ cres 5664  tpos ctpos 8221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-tpos 8222

This theorem is referenced by:  tposresg  49541