Theorem cutsf 27800

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 27768	. . . 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 7492	. . 3 ⊢ Fun |s
3		dmcuts 27799	. . 3 ⊢ dom |s = <<s
4		df-fn 6503	. . 3 ⊢ ( |s Fn <<s ↔ (Fun |s ∧ dom |s = <<s ))
5	2, 3, 4	mpbir2an 712	. 2 ⊢ |s Fn <<s
6	1	rnmpo 7501	. . 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 3446	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
8		vex 3446	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
9	7, 8	elimasn 6057	. . . . . . . . 9 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
10		df-br 5101	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
11	9, 10	bitr4i 278	. . . . . . . 8 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
12		cutsval 27788	. . . . . . . . 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 27790	. . . . . . . . 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 4022	. . 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 3982	. 2 ⊢ ran |s ⊆ No
22		df-f 6504	. 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 3401   ⊆ wss 3903  𝒫 cpw 4556  {csn 4582  ⟨cop 4588  ∩ cint 4904   class class class wbr 5100  dom cdm 5632  ran crn 5633   “ cima 5635  Fun wfun 6494   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  ℩crio 7324  (class class class)co 7368   No csur 27619   bday cbday 27621   <<s cslts 27765   |s ccuts 27767

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1o 8407  df-2o 8408  df-no 27622  df-lts 27623  df-bday 27624  df-slts 27766  df-cuts 27768

This theorem is referenced by:  madeval  27840  madeval2  27841  cutsfo  27913  madefi  27921