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Theorem scutf 33176
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 33156 . . . 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 7270 . . 3 Fun |s
3 dmscut 33175 . . 3 dom |s = <<s
4 df-fn 6357 . . 3 ( |s Fn <<s ↔ (Fun |s ∧ dom |s = <<s ))
52, 3, 4mpbir2an 707 . 2 |s Fn <<s
61rnmpo 7278 . . 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 3503 . . . . . . . . . 10 𝑎 ∈ V
8 vex 3503 . . . . . . . . . 10 𝑏 ∈ V
97, 8elimasn 5953 . . . . . . . . 9 (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
10 df-br 5064 . . . . . . . . 9 (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
119, 10bitr4i 279 . . . . . . . 8 (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
12 scutval 33168 . . . . . . . . 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 33169 . . . . . . . . . 10 (𝑎 <<s 𝑏 → ((𝑎 |s 𝑏) ∈ No 𝑎 <<s {(𝑎 |s 𝑏)} ∧ {(𝑎 |s 𝑏)} <<s 𝑏))
1413simp1d 1136 . . . . . . . . 9 (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No )
1512, 14eqeltrrd 2919 . . . . . . . 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 218 . . . . . . 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 2913 . . . . . . 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 482 . . . . 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 3297 . . . 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 4050 . . 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 4005 . 2 ran |s ⊆ No
23 df-f 6358 . 2 ( |s : <<s ⟶ No ↔ ( |s Fn <<s ∧ ran |s ⊆ No ))
245, 22, 23mpbir2an 707 1 |s : <<s ⟶ No
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  {cab 2804  wrex 3144  {crab 3147  wss 3940  𝒫 cpw 4542  {csn 4564  cop 4570   cint 4874   class class class wbr 5063  dom cdm 5554  ran crn 5555  cima 5557  Fun wfun 6348   Fn wfn 6349  wf 6350  cfv 6354  crio 7107  (class class class)co 7150   No csur 33050   bday cbday 33052   <<s csslt 33153   |s cscut 33155
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1o 8098  df-2o 8099  df-no 33053  df-slt 33054  df-bday 33055  df-sslt 33154  df-scut 33156
This theorem is referenced by:  madeval  33192  madeval2  33193
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