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Theorem dmscut 33276
 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 7258 . 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 33257 . . . 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-mpo 7164 . . . 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 2847 . . 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 5776 . 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 33255 . . . . 5 <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 No 𝑏 No ∧ ∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦)}
76relopabi 5697 . . . 4 Rel <<s
8 19.42v 1953 . . . . . 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 33261 . . . . . . . . 9 (𝑎 <<s 𝑏𝑎 No )
10 velpw 4547 . . . . . . . . 9 (𝑎 ∈ 𝒫 No 𝑎 No )
119, 10sylibr 236 . . . . . . . 8 (𝑎 <<s 𝑏𝑎 ∈ 𝒫 No )
1211pm4.71ri 563 . . . . . . 7 (𝑎 <<s 𝑏 ↔ (𝑎 ∈ 𝒫 No 𝑎 <<s 𝑏))
13 vex 3500 . . . . . . . . . 10 𝑎 ∈ V
14 vex 3500 . . . . . . . . . 10 𝑏 ∈ V
1513, 14elimasn 5957 . . . . . . . . 9 (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
16 df-br 5070 . . . . . . . . 9 (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
1715, 16bitr4i 280 . . . . . . . 8 (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
1817anbi2i 624 . . . . . . 7 ((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ↔ (𝑎 ∈ 𝒫 No 𝑎 <<s 𝑏))
19 riotaex 7121 . . . . . . . . 9 (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})) ∈ V
2019isseti 3511 . . . . . . . 8 𝑐 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))
2120biantru 532 . . . . . . 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)}))))
2212, 18, 213bitr2i 301 . . . . . 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)}))))
238, 22, 163bitr2ri 302 . . . . 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)}))))
2423a1i 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)})))))
257, 24opabbi2dv 5723 . . 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)})))})
2625mptru 1543 . 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)})))}
271, 5, 263eqtr4i 2857 1 dom |s = <<s
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536  ⊤wtru 1537  ∃wex 1779   ∈ wcel 2113  ∀wral 3141  {crab 3145   ⊆ wss 3939  𝒫 cpw 4542  {csn 4570  ⟨cop 4576  ∩ cint 4879   class class class wbr 5069  {copab 5131  dom cdm 5558   “ cima 5561  ‘cfv 6358  ℩crio 7116  {coprab 7160   ∈ cmpo 7161   No csur 33151
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