Theorem dmcuts 27799

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 7471	. 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 27768	. . . 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 7373	. . . 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 2760	. . 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 5861	. 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 27766	. . . . 5 ⊢ <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)}
7	6	relopabiv 5777	. . . 4 ⊢ Rel <<s
8		19.42v 1955	. . . . . 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 27773	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 → 𝑎 ⊆ No )
10		velpw 4561	. . . . . . . . 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 3446	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
14		vex 3446	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
15	13, 14	elimasn 6057	. . . . . . . . 9 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
16		df-br 5101	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
17	15, 16	bitr4i 278	. . . . . . . 8 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
18	17	anbi2i 624	. . . . . . 7 ⊢ ((𝑎 ∈ 𝒫 No ∧ 𝑏 ∈ ( <<s “ {𝑎})) ↔ (𝑎 ∈ 𝒫 No ∧ 𝑎 <<s 𝑏))
19		riotaex 7329	. . . . . . . . 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 3460	. . . . . . . 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 5806	. . 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 1549	. 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 2770	1 ⊢ dom |s = <<s

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ⊤wtru 1543  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  {crab 3401   ⊆ wss 3903  𝒫 cpw 4556  {csn 4582  ⟨cop 4588  ∩ cint 4904   class class class wbr 5100  {copab 5162  dom cdm 5632   “ cima 5635  ‘cfv 6500  ℩crio 7324  {coprab 7369   ∈ cmpo 7370   No csur 27619   <s clts 27620   bday cbday 27621   <<s cslts 27765   |s ccuts 27767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-riota 7325  df-oprab 7372  df-mpo 7373  df-slts 27766  df-cuts 27768

This theorem is referenced by:  cutsf  27800  madeval2  27841