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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   <s cslt 33152   bday cbday 33153   <<s csslt 33254   |s cscut 33256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-cnv 5566  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-riota 7117  df-oprab 7163  df-mpo 7164  df-sslt 33255  df-scut 33257
This theorem is referenced by:  scutf  33277  madeval2  33294
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