Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  scutf Structured version   Visualization version   GIF version

Theorem scutf 32225
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 32205 . . . 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 6927 . . 3 Fun |s
3 dmscut 32224 . . 3 dom |s = <<s
4 df-fn 6052 . . 3 ( |s Fn <<s ↔ (Fun |s ∧ dom |s = <<s ))
52, 3, 4mpbir2an 993 . 2 |s Fn <<s
61rnmpt2 6935 . . 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 3343 . . . . . . . . . 10 𝑎 ∈ V
8 vex 3343 . . . . . . . . . 10 𝑏 ∈ V
97, 8elimasn 5648 . . . . . . . . 9 (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
10 df-br 4805 . . . . . . . . 9 (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
119, 10bitr4i 267 . . . . . . . 8 (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
12 scutval 32217 . . . . . . . . 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 32218 . . . . . . . . . 10 (𝑎 <<s 𝑏 → ((𝑎 |s 𝑏) ∈ No 𝑎 <<s {(𝑎 |s 𝑏)} ∧ {(𝑎 |s 𝑏)} <<s 𝑏))
1413simp1d 1137 . . . . . . . . 9 (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No )
1512, 14eqeltrrd 2840 . . . . . . . 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 207 . . . . . . 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 2834 . . . . . . 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 473 . . . . 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 3174 . . . 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 3818 . . 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 3776 . 2 ran |s ⊆ No
23 df-f 6053 . 2 ( |s : <<s ⟶ No ↔ ( |s Fn <<s ∧ ran |s ⊆ No ))
245, 22, 23mpbir2an 993 1 |s : <<s ⟶ No
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {cab 2746  wrex 3051  {crab 3054  wss 3715  𝒫 cpw 4302  {csn 4321  cop 4327   cint 4627   class class class wbr 4804  dom cdm 5266  ran crn 5267  cima 5269  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049  crio 6773  (class class class)co 6813   No csur 32099   bday cbday 32101   <<s csslt 32202   |s cscut 32204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1o 7729  df-2o 7730  df-no 32102  df-slt 32103  df-bday 32104  df-sslt 32203  df-scut 32205
This theorem is referenced by:  madeval  32241  madeval2  32242
  Copyright terms: Public domain W3C validator