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Theorem eqscut 27689
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 27684 . . . . 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 4633 . . . . . . 7 (𝑥 = 𝑦 → {𝑥} = {𝑦})
43breq2d 5153 . . . . . 6 (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
53breq1d 5151 . . . . . 6 (𝑥 = 𝑦 → ({𝑥} <<s 𝑅 ↔ {𝑦} <<s 𝑅))
64, 5anbi12d 630 . . . . 5 (𝑥 = 𝑦 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
76riotarab 7403 . . . 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 2782 . . 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 2728 . 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 27683 . . . 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 3692 . . . 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 1086 . . . . . 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 4633 . . . . . . . 8 (𝑥 = 𝑋 → {𝑥} = {𝑋})
1514breq2d 5153 . . . . . . 7 (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
1614breq1d 5151 . . . . . . 7 (𝑥 = 𝑋 → ({𝑥} <<s 𝑅 ↔ {𝑋} <<s 𝑅))
17 fveqeq2 6893 . . . . . . 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 1432 . . . . . 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 7386 . . . 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 1084   = wceq 1533  wcel 2098  ∃!wreu 3368  {crab 3426  {csn 4623   cint 4943   class class class wbr 5141  cima 5672  cfv 6536  crio 7359  (class class class)co 7404   No csur 27524   bday cbday 27526   <<s csslt 27664   |s cscut 27666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1o 8464  df-2o 8465  df-no 27527  df-slt 27528  df-bday 27529  df-sslt 27665  df-scut 27667
This theorem is referenced by:  eqscut2  27690  cuteq0  27716  madebdaylemlrcut  27776  cofcut1  27791
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