Theorem scutval 33994

Description: The value of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
scutval	⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem scutval

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

Step	Hyp	Ref	Expression
1		ssltex1 33981	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
2		ssltss1 33983	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
3	1, 2	elpwd 4541	. 2 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ 𝒫 No )
4		df-br 5075	. . . 4 ⊢ (𝐴 <<s 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ <<s )
5	4	biimpi 215	. . 3 ⊢ (𝐴 <<s 𝐵 → ⟨𝐴, 𝐵⟩ ∈ <<s )
6		ssltex2 33982	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
7		elimasng 5996	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ ( <<s “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ <<s ))
8	1, 6, 7	syl2anc 584	. . 3 ⊢ (𝐴 <<s 𝐵 → (𝐵 ∈ ( <<s “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ <<s ))
9	5, 8	mpbird 256	. 2 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ ( <<s “ {𝐴}))
10		riotaex 7236	. . 3 ⊢ (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) ∈ V
11		breq1 5077	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 <<s {𝑦} ↔ 𝐴 <<s {𝑦}))
12		breq2 5078	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ({𝑦} <<s 𝑏 ↔ {𝑦} <<s 𝐵))
13	11, 12	bi2anan9 636	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏) ↔ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
14	13	rabbidv 3414	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} = {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
15	14	imaeq2d 5969	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}) = ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
16	15	inteqd 4884	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
17	16	eqeq2d 2749	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}) ↔ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
18	14, 17	riotaeqbidv 7235	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
19		sneq 4571	. . . . 5 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
20	19	imaeq2d 5969	. . . 4 ⊢ (𝑎 = 𝐴 → ( <<s “ {𝑎}) = ( <<s “ {𝐴}))
21		df-scut 33978	. . . 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)})))
22	18, 20, 21	ovmpox 7426	. . 3 ⊢ ((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)})) ∈ V) → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
23	10, 22	mp3an3 1449	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝐵 ∈ ( <<s “ {𝐴})) → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
24	3, 9, 23	syl2anc 584	1 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068  Vcvv 3432  𝒫 cpw 4533  {csn 4561  ⟨cop 4567  ∩ cint 4879   class class class wbr 5074   “ cima 5592  ‘cfv 6433  ℩crio 7231  (class class class)co 7275   No csur 33843   bday cbday 33845   <<s csslt 33975   |s cscut 33977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-sslt 33976  df-scut 33978

This theorem is referenced by:  scutcut  33995  scutbday  33998  eqscut  33999  scutun12  34004  scutf  34006  scutbdaylt  34012