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Theorem scutf 27791
Description: Functionality 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 27762 . . . 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)})))
21mpofun 7544 . . 3 Fun |s
3 dmscut 27790 . . 3 dom |s = <<s
4 df-fn 6552 . . 3 ( |s Fn <<s ↔ (Fun |s ∧ dom |s = <<s ))
52, 3, 4mpbir2an 709 . 2 |s Fn <<s
61rnmpo 7554 . . 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 3465 . . . . . . . . . 10 𝑎 ∈ V
8 vex 3465 . . . . . . . . . 10 𝑏 ∈ V
97, 8elimasn 6094 . . . . . . . . 9 (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
10 df-br 5150 . . . . . . . . 9 (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
119, 10bitr4i 277 . . . . . . . 8 (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
12 scutval 27779 . . . . . . . . 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 scutcl 27781 . . . . . . . . 9 (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No )
1412, 13eqeltrrd 2826 . . . . . . . 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 )
1511, 14sylbi 216 . . . . . . 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 )
16 eleq1a 2820 . . . . . . 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 ))
1715, 16syl 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 ))
1817adantl 480 . . . . 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 ))
1918rexlimivv 3189 . . . 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 )
2019abssi 4063 . . 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
216, 20eqsstri 4011 . 2 ran |s ⊆ No
22 df-f 6553 . 2 ( |s : <<s ⟶ No ↔ ( |s Fn <<s ∧ ran |s ⊆ No ))
235, 21, 22mpbir2an 709 1 |s : <<s ⟶ No
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {cab 2702  wrex 3059  {crab 3418  wss 3944  𝒫 cpw 4604  {csn 4630  cop 4636   cint 4950   class class class wbr 5149  dom cdm 5678  ran crn 5679  cima 5681  Fun wfun 6543   Fn wfn 6544  wf 6545  cfv 6549  crio 7374  (class class class)co 7419   No csur 27618   bday cbday 27620   <<s csslt 27759   |s cscut 27761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1o 8487  df-2o 8488  df-no 27621  df-slt 27622  df-bday 27623  df-sslt 27760  df-scut 27762
This theorem is referenced by:  madeval  27825  madeval2  27826  scutfo  27876
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