Theorem tposres0 48746

Description: The transposition of a set restricted to the empty set is the set restricted to the empty set. See also ressn 6272 and dftpos6 48744 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 5990	. 2 ⊢ Rel (tpos 𝐹 ↾ {∅})
2		relres 5990	. 2 ⊢ Rel (𝐹 ↾ {∅})
3		velsn 4615	. . . . 5 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4		brtpos0 8227	. . . . . . 7 ⊢ (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
5	4	elv 3462	. . . . . 6 ⊢ (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
6		breq1 5120	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
7		breq1 5120	. . . . . . 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 3461	. . . 4 ⊢ 𝑦 ∈ V
13	12	brresi 5973	. . 3 ⊢ (𝑥(tpos 𝐹 ↾ {∅})𝑦 ↔ (𝑥 ∈ {∅} ∧ 𝑥tpos 𝐹𝑦))
14	12	brresi 5973	. . 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 5768	1 ⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3457  ∅c0 4306  {csn 4599   class class class wbr 5117   ↾ cres 5654  tpos ctpos 8219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-fv 6536  df-tpos 8220

This theorem is referenced by:  tposresg  48747