Theorem eqcuts 28176

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 28171	. . . . 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 4590	. . . . . . 7 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
4	3	breq2d 5114	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
5	3	breq1d 5112	. . . . . 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 7419	. . . 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 2811	. . 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 2762	. 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 28170	. . . 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 3658	. . . 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 4590	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
15	14	breq2d 5114	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
16	14	breq1d 5112	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ({𝑥} <<s 𝑅 ↔ {𝑋} <<s 𝑅))
17		fveqeq2 6894	. . . . . . 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 7402	. . . 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 1569   ∈ wcel 2144  ∃!wreu 3363  {crab 3412  {csn 4584  ∩ cint 4906   class class class wbr 5102   “ cima 5653  ‘cfv 6538  ℩crio 7376  (class class class)co 7420  Nocsur 28002  bdaycbday 28004   <<s cslts 28148   |s ccuts 28150

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151

This theorem is used by:  eqcuts2  28177  cuteq0  28206  madebdaylemlrcut  28290  cofcut1  28311