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Theorem scutval 27161
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 27148 . . 3 (𝐴 <<s 𝐵𝐴 ∈ V)
2 ssltss1 27150 . . 3 (𝐴 <<s 𝐵𝐴 No )
31, 2elpwd 4571 . 2 (𝐴 <<s 𝐵𝐴 ∈ 𝒫 No )
4 df-br 5111 . . . 4 (𝐴 <<s 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ <<s )
54biimpi 215 . . 3 (𝐴 <<s 𝐵 → ⟨𝐴, 𝐵⟩ ∈ <<s )
6 ssltex2 27149 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
7 elimasng 6045 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ ( <<s “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ <<s ))
81, 6, 7syl2anc 585 . . 3 (𝐴 <<s 𝐵 → (𝐵 ∈ ( <<s “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ <<s ))
95, 8mpbird 257 . 2 (𝐴 <<s 𝐵𝐵 ∈ ( <<s “ {𝐴}))
10 riotaex 7322 . . 3 (𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) ∈ V
11 breq1 5113 . . . . . . 7 (𝑎 = 𝐴 → (𝑎 <<s {𝑦} ↔ 𝐴 <<s {𝑦}))
12 breq2 5114 . . . . . . 7 (𝑏 = 𝐵 → ({𝑦} <<s 𝑏 ↔ {𝑦} <<s 𝐵))
1311, 12bi2anan9 638 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏) ↔ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
1413rabbidv 3418 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} = {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
1514imaeq2d 6018 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
1615inteqd 4917 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
1716eqeq2d 2748 . . . . 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 7321 . . . 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 4601 . . . . 5 (𝑎 = 𝐴 → {𝑎} = {𝐴})
2019imaeq2d 6018 . . . 4 (𝑎 = 𝐴 → ( <<s “ {𝑎}) = ( <<s “ {𝐴}))
21 df-scut 27145 . . . 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 7513 . . 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 1451 . 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 585 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 397   = wceq 1542  wcel 2107  {crab 3410  Vcvv 3448  𝒫 cpw 4565  {csn 4591  cop 4597   cint 4912   class class class wbr 5110  cima 5641  cfv 6501  crio 7317  (class class class)co 7362   No csur 27004   bday cbday 27006   <<s csslt 27142   |s cscut 27144
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-sslt 27143  df-scut 27145
This theorem is referenced by:  scutcut  27162  scutbday  27165  eqscut  27166  scutun12  27171  scutf  27173  scutbdaylt  27179
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