Theorem cutsf 27798

Description: Functionality statement for the surreal cut operator. (Contributed by Scott Fenton, 15-Dec-2021.)

Assertion

Ref	Expression
cutsf	⊢ |s : <<s ⟶ No

Proof of Theorem cutsf

Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cuts 27766	. . . 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 7484	. . 3 ⊢ Fun |s
3		dmcuts 27797	. . 3 ⊢ dom |s = <<s
4		df-fn 6495	. . 3 ⊢ ( |s Fn <<s ↔ (Fun |s ∧ dom |s = <<s ))
5	2, 3, 4	mpbir2an 712	. 2 ⊢ |s Fn <<s
6	1	rnmpo 7493	. . 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 3434	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
8		vex 3434	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
9	7, 8	elimasn 6049	. . . . . . . . 9 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
10		df-br 5087	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
11	9, 10	bitr4i 278	. . . . . . . 8 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
12		cutsval 27786	. . . . . . . . 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		cutscl 27788	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No )
14	12, 13	eqeltrrd 2838	. . . . . . . 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 217	. . . . . . 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 2832	. . . . . . 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 481	. . . . 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 3180	. . . 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 4009	. . 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 3969	. 2 ⊢ ran |s ⊆ No
22		df-f 6496	. 2 ⊢ ( |s : <<s ⟶ No ↔ ( |s Fn <<s ∧ ran |s ⊆ No ))
23	5, 21, 22	mpbir2an 712	1 ⊢ |s : <<s ⟶ No

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062  {crab 3390   ⊆ wss 3890  𝒫 cpw 4542  {csn 4568  ⟨cop 4574  ∩ cint 4890   class class class wbr 5086  dom cdm 5624  ran crn 5625   “ cima 5627  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  ℩crio 7316  (class class class)co 7360   No csur 27617   bday cbday 27619   <<s cslts 27763   |s ccuts 27765

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1o 8398  df-2o 8399  df-no 27620  df-lts 27621  df-bday 27622  df-slts 27764  df-cuts 27766

This theorem is referenced by:  madeval  27838  madeval2  27839  cutsfo  27911  madefi  27919