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Theorem dmscut 27871
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 7535 . 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 27843 . . . 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 7436 . . . 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 2763 . . 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 5918 . 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 27841 . . . . 5 <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 No 𝑏 No ∧ ∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦)}
76relopabiv 5833 . . . 4 Rel <<s
8 19.42v 1951 . . . . . 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 27848 . . . . . . . . 9 (𝑎 <<s 𝑏𝑎 No )
10 velpw 4610 . . . . . . . . 9 (𝑎 ∈ 𝒫 No 𝑎 No )
119, 10sylibr 234 . . . . . . . 8 (𝑎 <<s 𝑏𝑎 ∈ 𝒫 No )
1211pm4.71ri 560 . . . . . . 7 (𝑎 <<s 𝑏 ↔ (𝑎 ∈ 𝒫 No 𝑎 <<s 𝑏))
13 vex 3482 . . . . . . . . . 10 𝑎 ∈ V
14 vex 3482 . . . . . . . . . 10 𝑏 ∈ V
1513, 14elimasn 6110 . . . . . . . . 9 (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
16 df-br 5149 . . . . . . . . 9 (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
1715, 16bitr4i 278 . . . . . . . 8 (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
1817anbi2i 623 . . . . . . 7 ((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) ↔ (𝑎 ∈ 𝒫 No 𝑎 <<s 𝑏))
19 riotaex 7392 . . . . . . . . 9 (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})) ∈ V
2019isseti 3496 . . . . . . . 8 𝑐 𝑐 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))
2120biantru 529 . . . . . . 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 299 . . . . . 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 300 . . . . 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 5863 . . 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 1544 . 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 2773 1 dom |s = <<s
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wtru 1538  wex 1776  wcel 2106  wral 3059  {crab 3433  wss 3963  𝒫 cpw 4605  {csn 4631  cop 4637   cint 4951   class class class wbr 5148  {copab 5210  dom cdm 5689  cima 5692  cfv 6563  crio 7387  {coprab 7432  cmpo 7433   No csur 27699   <s cslt 27700   bday cbday 27701   <<s csslt 27840   |s cscut 27842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-riota 7388  df-oprab 7435  df-mpo 7436  df-sslt 27841  df-scut 27843
This theorem is referenced by:  scutf  27872  madeval2  27907
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