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Theorem eqscut 27751
Description: Condition for equality to a surreal cut. (Contributed by Scott Fenton, 8-Aug-2024.)
Assertion
Ref Expression
eqscut ((𝐿 <<s 𝑅𝑋 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.
StepHypRef 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)})))
21adantr 480 . . . 4 ((𝐿 <<s 𝑅𝑋 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 4639 . . . . . . 7 (𝑥 = 𝑦 → {𝑥} = {𝑦})
43breq2d 5160 . . . . . 6 (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
53breq1d 5158 . . . . . 6 (𝑥 = 𝑦 → ({𝑥} <<s 𝑅 ↔ {𝑦} <<s 𝑅))
64, 5anbi12d 631 . . . . 5 (𝑥 = 𝑦 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
76riotarab 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)})))
82, 7eqtrdi 2784 . . 3 ((𝐿 <<s 𝑅𝑋 No ) → (𝐿 |s 𝑅) = (𝑥 No ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ∧ ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))))
98eqeq1d 2730 . 2 ((𝐿 <<s 𝑅𝑋 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)}))
116reurab 3696 . . . 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)})))
1210, 11sylib 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 4639 . . . . . . . 8 (𝑥 = 𝑋 → {𝑥} = {𝑋})
1514breq2d 5160 . . . . . . 7 (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
1614breq1d 5158 . . . . . . 7 (𝑥 = 𝑋 → ({𝑥} <<s 𝑅 ↔ {𝑋} <<s 𝑅))
17 fveqeq2 6906 . . . . . . 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)})))
1815, 16, 173anbi123d 1433 . . . . . 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)}))))
1913, 18bitr3id 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)}))))
2019riota2 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))
2120ancoms 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))
2212, 21sylan 579 . 2 ((𝐿 <<s 𝑅𝑋 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))
239, 22bitr4d 282 1 ((𝐿 <<s 𝑅𝑋 No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday 𝑋) = ( bday “ {𝑦 No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  ∃!wreu 3371  {crab 3429  {csn 4629   cint 4949   class class class wbr 5148  cima 5681  cfv 6548  crio 7375  (class class class)co 7420   No csur 27586   bday cbday 27588   <<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 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1o 8487  df-2o 8488  df-no 27589  df-slt 27590  df-bday 27591  df-sslt 27727  df-scut 27729
This theorem is referenced by:  eqscut2  27752  cuteq0  27778  madebdaylemlrcut  27838  cofcut1  27853
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