Theorem dmcuts 28064

Description: The domain of the surreal cut operation is all separated surreal sets. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
dmcuts	⊢ dom |s = <<s

Proof of Theorem dmcuts

Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmoprab 7520	. 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-cuts 28033	. . . 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 7422	. . . 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 2785	. . 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 5892	. 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-slts 28031	. . . . 5 ⊢ <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)}
7	6	relopabiv 5805	. . . 4 ⊢ Rel <<s
8		19.42v 1986	. . . . . 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		sltsss1 28038	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 → 𝑎 ⊆ No )
10		velpw 4565	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 No ↔ 𝑎 ⊆ No )
11	9, 10	sylibr 237	. . . . . . . 8 ⊢ (𝑎 <<s 𝑏 → 𝑎 ∈ 𝒫 No )
12	11	pm4.71ri 570	. . . . . . 7 ⊢ (𝑎 <<s 𝑏 ↔ (𝑎 ∈ 𝒫 No ∧ 𝑎 <<s 𝑏))
13		vex 3457	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
14		vex 3457	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
15	13, 14	elimasn 6090	. . . . . . . . 9 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
16		df-br 5108	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
17	15, 16	bitr4i 281	. . . . . . . 8 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
18	17	anbi2i 635	. . . . . . 7 ⊢ ((𝑎 ∈ 𝒫 No ∧ 𝑏 ∈ ( <<s “ {𝑎})) ↔ (𝑎 ∈ 𝒫 No ∧ 𝑎 <<s 𝑏))
19		riotaex 7378	. . . . . . . . 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 3471	. . . . . . . 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 539	. . . . . . 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 302	. . . . . 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 303	. . . . 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 5833	. . 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 1577	. 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 2795	1 ⊢ dom |s = <<s

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571  ∃wex 1812   ∈ wcel 2145  ∀wral 3078  {crab 3414   ⊆ wss 3902  𝒫 cpw 4560  {csn 4587  ⟨cop 4593  ∩ cint 4910   class class class wbr 5107  {copab 5171  dom cdm 5659   “ cima 5662  ‘cfv 6537  ℩crio 7373  {coprab 7418   ∈ cmpo 7419   No csur 27884   <s clts 27885   bday cbday 27886   <<s cslts 28030   |s ccuts 28032

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-riota 7374  df-oprab 7421  df-mpo 7422  df-slts 28031  df-cuts 28033

This theorem is used by:  cutsf  28065  madeval2  28106