Theorem scutf 31919

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.

Step	Hyp	Ref	Expression
1		df-scut 31899	. . . 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	mpt2fun 6762	. . 3 ⊢ Fun |s
3		dmscut 31918	. . 3 ⊢ dom |s = <<s
4		df-fn 5891	. . 3 ⊢ ( |s Fn <<s ↔ (Fun |s ∧ dom |s = <<s ))
5	2, 3, 4	mpbir2an 955	. 2 ⊢ |s Fn <<s
6	1	rnmpt2 6770	. . 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 3203	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
8		vex 3203	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
9	7, 8	elimasn 5490	. . . . . . . . 9 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
10		df-br 4654	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
11	9, 10	bitr4i 267	. . . . . . . 8 ⊢ (𝑏 ∈ ( <<s “ {𝑎}) ↔ 𝑎 <<s 𝑏)
12		scutval 31911	. . . . . . . . 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 31912	. . . . . . . . . 10 ⊢ (𝑎 <<s 𝑏 → ((𝑎 |s 𝑏) ∈ No ∧ 𝑎 <<s {(𝑎 |s 𝑏)} ∧ {(𝑎 |s 𝑏)} <<s 𝑏))
14	13	simp1d 1073	. . . . . . . . 9 ⊢ (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No )
15	12, 14	eqeltrrd 2702	. . . . . . . 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 )
16	11, 15	sylbi 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 2696	. . . . . . 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 ))
18	16, 17	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 ))
19	18	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 ))
20	19	rexlimivv 3036	. . . 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 )
21	20	abssi 3677	. . 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
22	6, 21	eqsstri 3635	. 2 ⊢ ran |s ⊆ No
23		df-f 5892	. 2 ⊢ ( |s : <<s ⟶ No ↔ ( |s Fn <<s ∧ ran |s ⊆ No ))
24	5, 22, 23	mpbir2an 955	1 ⊢ |s : <<s ⟶ No

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∃wrex 2913  {crab 2916   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  ⟨cop 4183  ∩ cint 4475   class class class wbr 4653  dom cdm 5114  ran crn 5115   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  ℩crio 6610  (class class class)co 6650   No csur 31793   bday cbday 31795   <<s csslt 31896   |s cscut 31898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797  df-bday 31798  df-sslt 31897  df-scut 31899

This theorem is referenced by:  madeval  31935  madeval2  31936