Theorem dmcuts 27861

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 7495	. 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 27830	. . . 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 7397	. . . 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 2784	. . 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 5878	. 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 27828	. . . . 5 ⊢ <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)}
7	6	relopabiv 5791	. . . 4 ⊢ Rel <<s
8		19.42v 1972	. . . . . 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 27835	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 → 𝑎 ⊆ No )
10		velpw 4559	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 No ↔ 𝑎 ⊆ No )
11	9, 10	sylibr 236	. . . . . . . 8 ⊢ (𝑎 <<s 𝑏 → 𝑎 ∈ 𝒫 No )
12	11	pm4.71ri 568	. . . . . . 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 6076	. . . . . . . . 9 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
16		df-br 5100	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
17	15, 16	bitr4i 280	. . . . . . . 8 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
18	17	anbi2i 632	. . . . . . 7 ⊢ ((𝑎 ∈ 𝒫 No ∧ 𝑏 ∈ ( <<s “ {𝑎})) ↔ (𝑎 ∈ 𝒫 No ∧ 𝑎 <<s 𝑏))
19		riotaex 7353	. . . . . . . . 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 537	. . . . . . 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 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)}))))
23	8, 22, 16	3bitr2ri 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)}))))
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 5819	. . 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 1566	. 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 2794	1 ⊢ dom |s = <<s

Syntax hints:   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559  ⊤wtru 1560  ∃wex 1798   ∈ wcel 2141  ∀wral 3075  {crab 3413   ⊆ wss 3904  𝒫 cpw 4554  {csn 4581  ⟨cop 4587  ∩ cint 4904   class class class wbr 5099  {copab 5161  dom cdm 5645   “ cima 5648  ‘cfv 6517  ℩crio 7348  {coprab 7393   ∈ cmpo 7394   No csur 27681   <s clts 27682   bday cbday 27683   <<s cslts 27827   |s ccuts 27829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-riota 7349  df-oprab 7396  df-mpo 7397  df-slts 27828  df-cuts 27830

This theorem is referenced by:  cutsf  27862  madeval2  27903