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Theorem scutf 32234
Description: Functionhood statement for the surreal cut operator. (Contributed by Scott Fenton, 15-Dec-2021.)
Assertion
Ref Expression
scutf |s : <<s ⟶ No

Proof of Theorem scutf
Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-scut 32214 . . . 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)})))
21mpt2fun 6986 . . 3 Fun |s
3 dmscut 32233 . . 3 dom |s = <<s
4 df-fn 6098 . . 3 ( |s Fn <<s ↔ (Fun |s ∧ dom |s = <<s ))
52, 3, 4mpbir2an 693 . 2 |s Fn <<s
61rnmpt2 6994 . . 3 ran |s = {𝑧 ∣ ∃𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})𝑧 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))}
7 vex 3390 . . . . . . . . . 10 𝑎 ∈ V
8 vex 3390 . . . . . . . . . 10 𝑏 ∈ V
97, 8elimasn 5694 . . . . . . . . 9 (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
10 df-br 4838 . . . . . . . . 9 (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
119, 10bitr4i 269 . . . . . . . 8 (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
12 scutval 32226 . . . . . . . . 9 (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)})))
13 scutcut 32227 . . . . . . . . . 10 (𝑎 <<s 𝑏 → ((𝑎 |s 𝑏) ∈ No 𝑎 <<s {(𝑎 |s 𝑏)} ∧ {(𝑎 |s 𝑏)} <<s 𝑏))
1413simp1d 1165 . . . . . . . . 9 (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No )
1512, 14eqeltrrd 2882 . . . . . . . 8 (𝑎 <<s 𝑏 → (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})) ∈ No )
1611, 15sylbi 208 . . . . . . 7 (𝑏 ∈ ( <<s “ {𝑎}) → (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})) ∈ No )
17 eleq1a 2876 . . . . . . 7 ((𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})) ∈ No → (𝑧 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})) → 𝑧 No ))
1816, 17syl 17 . . . . . 6 (𝑏 ∈ ( <<s “ {𝑎}) → (𝑧 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})) → 𝑧 No ))
1918adantl 469 . . . . 5 ((𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})) → (𝑧 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})) → 𝑧 No ))
2019rexlimivv 3220 . . . 4 (∃𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})𝑧 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})) → 𝑧 No )
2120abssi 3868 . . 3 {𝑧 ∣ ∃𝑎 ∈ 𝒫 No 𝑏 ∈ ( <<s “ {𝑎})𝑧 = (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))} ⊆ No
226, 21eqsstri 3826 . 2 ran |s ⊆ No
23 df-f 6099 . 2 ( |s : <<s ⟶ No ↔ ( |s Fn <<s ∧ ran |s ⊆ No ))
245, 22, 23mpbir2an 693 1 |s : <<s ⟶ No
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2155  {cab 2788  wrex 3093  {crab 3096  wss 3763  𝒫 cpw 4345  {csn 4364  cop 4370   cint 4662   class class class wbr 4837  dom cdm 5305  ran crn 5306  cima 5308  Fun wfun 6089   Fn wfn 6090  wf 6091  cfv 6095  crio 6828  (class class class)co 6868   No csur 32108   bday cbday 32110   <<s csslt 32211   |s cscut 32213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rmo 3100  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-int 4663  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-ord 5933  df-on 5934  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-riota 6829  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-1o 7790  df-2o 7791  df-no 32111  df-slt 32112  df-bday 32113  df-sslt 32212  df-scut 32214
This theorem is referenced by:  madeval  32250  madeval2  32251
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