Theorem eqscut 27751

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 27746	. . . . 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 4595	. . . . . . 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 632	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
7	6	riotarab 7368	. . . 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 2780	. . 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 2731	. 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 27745	. . . 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 3669	. . . 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 4595	. . . . . . . 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 6849	. . . . . . 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 1438	. . . . . 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 7351	. . . 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 1540   ∈ wcel 2109  ∃!wreu 3349  {crab 3402  {csn 4585  ∩ cint 4906   class class class wbr 5102   “ cima 5634  ‘cfv 6499  ℩crio 7325  (class class class)co 7369   No csur 27584   bday cbday 27586   <<s csslt 27726   |s cscut 27728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1o 8411  df-2o 8412  df-no 27587  df-slt 27588  df-bday 27589  df-sslt 27727  df-scut 27729

This theorem is referenced by:  eqscut2  27752  cuteq0  27781  madebdaylemlrcut  27848  cofcut1  27868