Theorem scutf 27302

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.

Step	Hyp	Ref	Expression
1		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)})))
2	1	mpofun 7528	. . 3 ⊢ Fun |s
3		dmscut 27301	. . 3 ⊢ dom |s = <<s
4		df-fn 6543	. . 3 ⊢ ( |s Fn <<s ↔ (Fun |s ∧ dom |s = <<s ))
5	2, 3, 4	mpbir2an 709	. 2 ⊢ |s Fn <<s
6	1	rnmpo 7538	. . 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 3478	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
8		vex 3478	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
9	7, 8	elimasn 6085	. . . . . . . . 9 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
10		df-br 5148	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
11	9, 10	bitr4i 277	. . . . . . . 8 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
12		scutval 27290	. . . . . . . . 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 27292	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No )
14	12, 13	eqeltrrd 2834	. . . . . . . 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 )
15	11, 14	sylbi 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 2828	. . . . . . 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 ))
17	15, 16	syl 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 ))
18	17	adantl 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 ))
19	18	rexlimivv 3199	. . . 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 )
20	19	abssi 4066	. . 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
21	6, 20	eqsstri 4015	. 2 ⊢ ran |s ⊆ No
22		df-f 6544	. 2 ⊢ ( |s : <<s ⟶ No ↔ ( |s Fn <<s ∧ ran |s ⊆ No ))
23	5, 21, 22	mpbir2an 709	1 ⊢ |s : <<s ⟶ No

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  ∃wrex 3070  {crab 3432   ⊆ wss 3947  𝒫 cpw 4601  {csn 4627  ⟨cop 4633  ∩ cint 4949   class class class wbr 5147  dom cdm 5675  ran crn 5676   “ cima 5678  Fun wfun 6534   Fn wfn 6535  ⟶wf 6536  ‘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-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  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-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  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-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1o 8462  df-2o 8463  df-no 27135  df-slt 27136  df-bday 27137  df-sslt 27272  df-scut 27274

This theorem is referenced by:  madeval  27336  madeval2  27337  scutfo  27387