Theorem eqscut 27865

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

Assertion

Ref	Expression
eqscut	⊢ ((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 eqscut

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		scutval 27860	. . . . 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 480	. . . 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 4641	. . . . . . 7 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
4	3	breq2d 5160	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
5	3	breq1d 5158	. . . . . 6 ⊢ (𝑥 = 𝑦 → ({𝑥} <<s 𝑅 ↔ {𝑦} <<s 𝑅))
6	4, 5	anbi12d 632	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
7	6	riotarab 7430	. . . 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 2791	. . 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 2737	. 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 27859	. . . 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 3710	. . . 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 218	. . 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 1088	. . . . . 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 4641	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
15	14	breq2d 5160	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
16	14	breq1d 5158	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ({𝑥} <<s 𝑅 ↔ {𝑋} <<s 𝑅))
17		fveqeq2 6916	. . . . . . 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 1435	. . . . . 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 285	. . . . 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 7413	. . . 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 458	. . 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 580	. 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 282	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)}))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∃!wreu 3376  {crab 3433  {csn 4631  ∩ cint 4951   class class class wbr 5148   “ cima 5692  ‘cfv 6563  ℩crio 7387  (class class class)co 7431   No csur 27699   bday cbday 27701   <<s csslt 27840   |s cscut 27842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843

This theorem is referenced by:  eqscut2  27866  cuteq0  27892  madebdaylemlrcut  27952  cofcut1  27969