Theorem tposres0 49806

Description: The transposition of a set restricted to the empty set is the set restricted to the empty set. See also ressn 6283 and dftpos6 49804 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 5998	. 2 ⊢ Rel (tpos 𝐹 ↾ {∅})
2		relres 5998	. 2 ⊢ Rel (𝐹 ↾ {∅})
3		velsn 4600	. . . . 5 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4		brtpos0 8232	. . . . . . 7 ⊢ (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
5	4	elv 3455	. . . . . 6 ⊢ (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
6		breq1 5106	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
7		breq1 5106	. . . . . . 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 585	. . 3 ⊢ ((𝑥 ∈ {∅} ∧ 𝑥tpos 𝐹𝑦) ↔ (𝑥 ∈ {∅} ∧ 𝑥𝐹𝑦))
12		vex 3454	. . . 4 ⊢ 𝑦 ∈ V
13	12	brresi 5981	. . 3 ⊢ (𝑥(tpos 𝐹 ↾ {∅})𝑦 ↔ (𝑥 ∈ {∅} ∧ 𝑥tpos 𝐹𝑦))
14	12	brresi 5981	. . 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 5771	1 ⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ∅c0 4279  {csn 4584   class class class wbr 5103   ↾ cres 5657  tpos ctpos 8224

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541  df-tpos 8225

This theorem is used by:  tposresg  49807