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Theorem scutval 27290
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 27277 . . 3 (𝐴 <<s 𝐵𝐴 ∈ V)
2 ssltss1 27279 . . 3 (𝐴 <<s 𝐵𝐴 No )
31, 2elpwd 4607 . 2 (𝐴 <<s 𝐵𝐴 ∈ 𝒫 No )
4 df-br 5148 . . . 4 (𝐴 <<s 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ <<s )
54biimpi 215 . . 3 (𝐴 <<s 𝐵 → ⟨𝐴, 𝐵⟩ ∈ <<s )
6 ssltex2 27278 . . . 4 (𝐴 <<s 𝐵𝐵 ∈ V)
7 elimasng 6084 . . . 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 7365 . . 3 (𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) ∈ V
11 breq1 5150 . . . . . . 7 (𝑎 = 𝐴 → (𝑎 <<s {𝑦} ↔ 𝐴 <<s {𝑦}))
12 breq2 5151 . . . . . . 7 (𝑏 = 𝐵 → ({𝑦} <<s 𝑏 ↔ {𝑦} <<s 𝐵))
1311, 12bi2anan9 637 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏) ↔ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
1413rabbidv 3440 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} = {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
1514imaeq2d 6057 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
1615inteqd 4954 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
1716eqeq2d 2743 . . . . 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 7364 . . . 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 4637 . . . . 5 (𝑎 = 𝐴 → {𝑎} = {𝐴})
2019imaeq2d 6057 . . . 4 (𝑎 = 𝐴 → ( <<s “ {𝑎}) = ( <<s “ {𝐴}))
21 df-scut 27274 . . . 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 7557 . . 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 1450 . 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 1541  wcel 2106  {crab 3432  Vcvv 3474  𝒫 cpw 4601  {csn 4627  cop 4633   cint 4949   class class class wbr 5147  cima 5678  cfv 6540  crio 7360  (class class class)co 7405   No csur 27132   bday cbday 27134   <<s csslt 27271   |s cscut 27273
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-sslt 27272  df-scut 27274
This theorem is referenced by:  scutcut  27291  scutbday  27294  eqscut  27295  scutun12  27300  scutf  27302  scutbdaylt  27308
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