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Theorem dmscut 32043
 Description: The domain of the surreal cut operation is all separated surreal sets. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
dmscut dom |s = <<s

Proof of Theorem dmscut
Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmoprab 6783 . 2 dom {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ∧ 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ∧ 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))}
2 df-scut 32024 . . . 4 |s = (𝑎 ∈ 𝒫 No , 𝑏 ∈ ( <<s “ {𝑎}) ↦ (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))
3 df-mpt2 6695 . . . 4 (𝑎 ∈ 𝒫 No , 𝑏 ∈ ( <<s “ {𝑎}) ↦ (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))) = {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ∧ 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))}
42, 3eqtri 2673 . . 3 |s = {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ∧ 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))}
54dmeqi 5357 . 2 dom |s = dom {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ∧ 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))}
6 df-sslt 32022 . . . . 5 <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 No 𝑏 No ∧ ∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦)}
76relopabi 5278 . . . 4 Rel <<s
8 19.42v 1921 . . . . . 6 (∃𝑐((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ∧ 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))) ↔ ((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ∧ ∃𝑐 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))))
9 ssltss1 32028 . . . . . . . . 9 (𝑎 <<s 𝑏𝑎 No )
10 vex 3234 . . . . . . . . . 10 𝑎 ∈ V
1110elpw 4197 . . . . . . . . 9 (𝑎 ∈ 𝒫 No 𝑎 No )
129, 11sylibr 224 . . . . . . . 8 (𝑎 <<s 𝑏𝑎 ∈ 𝒫 No )
1312pm4.71ri 666 . . . . . . 7 (𝑎 <<s 𝑏 ↔ (𝑎 ∈ 𝒫 No 𝑎 <<s 𝑏))
14 vex 3234 . . . . . . . . . 10 𝑏 ∈ V
1510, 14elimasn 5525 . . . . . . . . 9 (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
16 df-br 4686 . . . . . . . . 9 (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
1715, 16bitr4i 267 . . . . . . . 8 (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
1817anbi2i 730 . . . . . . 7 ((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ↔ (𝑎 ∈ 𝒫 No 𝑎 <<s 𝑏))
19 riotaex 6655 . . . . . . . . 9 (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})) ∈ V
20 isset 3238 . . . . . . . . 9 ((𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})) ∈ V ↔ ∃𝑐 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))
2119, 20mpbi 220 . . . . . . . 8 𝑐 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))
2221biantru 525 . . . . . . 7 ((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ↔ ((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ∧ ∃𝑐 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))))
2313, 18, 223bitr2i 288 . . . . . 6 (𝑎 <<s 𝑏 ↔ ((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ∧ ∃𝑐 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))))
248, 23, 163bitr2ri 289 . . . . 5 (⟨𝑎, 𝑏⟩ ∈ <<s ↔ ∃𝑐((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ∧ 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))))
2524a1i 11 . . . 4 (⊤ → (⟨𝑎, 𝑏⟩ ∈ <<s ↔ ∃𝑐((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ∧ 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))))
267, 25opabbi2dv 5304 . . 3 (⊤ → <<s = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ∧ 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))})
2726trud 1533 . 2 <<s = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ∧ 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))}
281, 5, 273eqtr4i 2683 1 dom |s = <<s
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ⊤wtru 1524  ∃wex 1744   ∈ wcel 2030  ∀wral 2941  {crab 2945  Vcvv 3231   ⊆ wss 3607  𝒫 cpw 4191  {csn 4210  ⟨cop 4216  ∩ cint 4507   class class class wbr 4685  {copab 4745  dom cdm 5143   “ cima 5146  ‘cfv 5926  ℩crio 6650  {coprab 6691   ↦ cmpt2 6692   No csur 31918
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