Theorem eqscut 33926

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 33921	. . . . 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 4568	. . . . . . 7 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
4	3	breq2d 5082	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
5	3	breq1d 5080	. . . . . 6 ⊢ (𝑥 = 𝑦 → ({𝑥} <<s 𝑅 ↔ {𝑦} <<s 𝑅))
6	4, 5	anbi12d 630	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
7	6	riotarab 33575	. . . 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 2795	. . 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 2740	. 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 33920	. . . 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 33576	. . . 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 217	. . 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 1087	. . . . . 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 4568	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
15	14	breq2d 5082	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
16	14	breq1d 5080	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ({𝑥} <<s 𝑅 ↔ {𝑋} <<s 𝑅))
17		fveqeq2 6765	. . . . . . 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 1434	. . . . . 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 284	. . . . 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 7238	. . . 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 579	. 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 281	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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∃!wreu 3065  {crab 3067  {csn 4558  ∩ cint 4876   class class class wbr 5070   “ cima 5583  ‘cfv 6418  ℩crio 7211  (class class class)co 7255   No csur 33770   bday cbday 33772   <<s csslt 33902   |s cscut 33904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-bday 33775  df-sslt 33903  df-scut 33905

This theorem is referenced by:  eqscut2  33927  madebdaylemlrcut  34006  cofcut1  34017