Theorem eqcuts 28058

Description: Condition for equality to a surreal cut. (Contributed by Scott Fenton, 8-Aug-2024.)

Assertion

Ref	Expression
eqcuts	⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))))

Distinct variable groups:   𝑦,𝐿   𝑦,𝑅

Allowed substitution hint:   𝑋(𝑦)

Proof of Theorem eqcuts

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cutsval 28053	. . . . 5 ⊢ (𝐿 <<s 𝑅 → (𝐿 |s 𝑅) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})))
2	1	adantr 486	. . . 4 ⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) → (𝐿 |s 𝑅) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})))
3		sneq 4597	. . . . . . 7 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
4	3	breq2d 5119	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
5	3	breq1d 5117	. . . . . 6 ⊢ (𝑥 = 𝑦 → ({𝑥} <<s 𝑅 ↔ {𝑦} <<s 𝑅))
6	4, 5	anbi12d 644	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
7	6	riotarab 7416	. . . 4 ⊢ (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})) = (℩𝑥 ∈ No ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ∧ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})))
8	2, 7	eqtrdi 2813	. . 3 ⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) → (𝐿 |s 𝑅) = (℩𝑥 ∈ No ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ∧ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))))
9	8	eqeq1d 2764	. 2 ⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (℩𝑥 ∈ No ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ∧ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))) = 𝑋))
10		conway 28052	. . . 4 ⊢ (𝐿 <<s 𝑅 → ∃!𝑥 ∈ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))
11	6	reurab 3662	. . . 4 ⊢ (∃!𝑥 ∈ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}) ↔ ∃!𝑥 ∈ No ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ∧ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})))
12	10, 11	sylib 221	. . 3 ⊢ (𝐿 <<s 𝑅 → ∃!𝑥 ∈ No ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ∧ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})))
13		df-3an 1105	. . . . . 6 ⊢ ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅 ∧ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})) ↔ ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ∧ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})))
14		sneq 4597	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
15	14	breq2d 5119	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
16	14	breq1d 5117	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ({𝑥} <<s 𝑅 ↔ {𝑋} <<s 𝑅))
17		fveqeq2 6891	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}) ↔ ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})))
18	15, 16, 17	3anbi123d 1464	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅 ∧ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})) ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))))
19	13, 18	bitr3id 288	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ∧ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})) ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))))
20	19	riota2 7399	. . . 4 ⊢ ((𝑋 ∈ No ∧ ∃!𝑥 ∈ No ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ∧ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))) → ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})) ↔ (℩𝑥 ∈ No ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ∧ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))) = 𝑋))
21	20	ancoms 464	. . 3 ⊢ ((∃!𝑥 ∈ No ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ∧ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})) ∧ 𝑋 ∈ No ) → ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})) ↔ (℩𝑥 ∈ No ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ∧ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))) = 𝑋))
22	12, 21	sylan 592	. 2 ⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) → ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)})) ↔ (℩𝑥 ∈ No ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ∧ ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))) = 𝑋))
23	9, 22	bitr4d 285	1 ⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃!wreu 3365  {crab 3414  {csn 4587  ∩ cint 4910   class class class wbr 5107   “ cima 5662  ‘cfv 6537  ℩crio 7373  (class class class)co 7417   No csur 27884   bday cbday 27886   <<s cslts 28030   |s ccuts 28032

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1o 8459  df-2o 8460  df-no 27887  df-lts 27888  df-bday 27889  df-slts 28031  df-cuts 28033

This theorem is used by:  eqcuts2  28059  cuteq0  28088  madebdaylemlrcut  28172  cofcut1  28193