Theorem dmscut 27856

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.

Step	Hyp	Ref	Expression
1		dmoprab 7536	. 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 27828	. . . 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)})))}
4	2, 3	eqtri 2765	. . 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)})))}
5	4	dmeqi 5915	. 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 27826	. . . . 5 ⊢ <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)}
7	6	relopabiv 5830	. . . 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 27833	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 → 𝑎 ⊆ No )
10		velpw 4605	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 No ↔ 𝑎 ⊆ No )
11	9, 10	sylibr 234	. . . . . . . 8 ⊢ (𝑎 <<s 𝑏 → 𝑎 ∈ 𝒫 No )
12	11	pm4.71ri 560	. . . . . . 7 ⊢ (𝑎 <<s 𝑏 ↔ (𝑎 ∈ 𝒫 No ∧ 𝑎 <<s 𝑏))
13		vex 3484	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
14		vex 3484	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
15	13, 14	elimasn 6108	. . . . . . . . 9 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
16		df-br 5144	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
17	15, 16	bitr4i 278	. . . . . . . 8 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
18	17	anbi2i 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
20	19	isseti 3498	. . . . . . . 8 ⊢ ∃𝑐 𝑐 = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))
21	20	biantru 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)}))))
22	12, 18, 21	3bitr2i 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)}))))
23	8, 22, 16	3bitr2ri 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)}))))
24	23	a1i 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)})))))
25	7, 24	opabbi2dv 5860	. . 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)})))})
26	25	mptru 1547	. 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)})))}
27	1, 5, 26	3eqtr4i 2775	1 ⊢ dom |s = <<s

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540  ⊤wtru 1541  ∃wex 1779   ∈ wcel 2108  ∀wral 3061  {crab 3436   ⊆ wss 3951  𝒫 cpw 4600  {csn 4626  ⟨cop 4632  ∩ cint 4946   class class class wbr 5143  {copab 5205  dom cdm 5685   “ cima 5688  ‘cfv 6561  ℩crio 7387  {coprab 7432   ∈ cmpo 7433   No csur 27684   <s cslt 27685   bday cbday 27686   <<s csslt 27825   |s cscut 27827

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-riota 7388  df-oprab 7435  df-mpo 7436  df-sslt 27826  df-scut 27828

This theorem is referenced by:  scutf  27857  madeval2  27892