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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.
StepHypRef Expression
1 ssltex1 33981 . . 3 (𝐴 <<s 𝐵𝐴 ∈ V)
2 ssltss1 33983 . . 3 (𝐴 <<s 𝐵𝐴 No )
31, 2elpwd 4541 . 2 (𝐴 <<s 𝐵𝐴 ∈ 𝒫 No )
4 df-br 5075 . . . 4 (𝐴 <<s 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ <<s )
54biimpi 215 . . 3 (𝐴 <<s 𝐵 → ⟨𝐴, 𝐵⟩ ∈ <<s )
6 ssltex2 33982 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
7 elimasng 5996 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ ( <<s “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ <<s ))
81, 6, 7syl2anc 584 . . 3 (𝐴 <<s 𝐵 → (𝐵 ∈ ( <<s “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ <<s ))
95, 8mpbird 256 . 2 (𝐴 <<s 𝐵𝐵 ∈ ( <<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))
1311, 12bi2anan9 636 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏) ↔ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
1413rabbidv 3414 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} = {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
1514imaeq2d 5969 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
1615inteqd 4884 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
1716eqeq2d 2749 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}) ↔ ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
1814, 17riotaeqbidv 7235 . . . 4 ((𝑎 = 𝐴𝑏 = 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})
2019imaeq2d 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)})))
2218, 20, 21ovmpox 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)})))
2310, 22mp3an3 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)})))
243, 9, 23syl2anc 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)})))
Colors of variables: wff setvar class
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
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