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Theorem eqscut 27166
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 27161 . . . . 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 482 . . . 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 4597 . . . . . . 7 (𝑥 = 𝑦 → {𝑥} = {𝑦})
43breq2d 5118 . . . . . 6 (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
53breq1d 5116 . . . . . 6 (𝑥 = 𝑦 → ({𝑥} <<s 𝑅 ↔ {𝑦} <<s 𝑅))
64, 5anbi12d 632 . . . . 5 (𝑥 = 𝑦 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
76riotarab 7357 . . . 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 2789 . . 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 2735 . 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 27160 . . . 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 3660 . . . 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 1090 . . . . . 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 4597 . . . . . . . 8 (𝑥 = 𝑋 → {𝑥} = {𝑋})
1514breq2d 5118 . . . . . . 7 (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
1614breq1d 5116 . . . . . . 7 (𝑥 = 𝑋 → ({𝑥} <<s 𝑅 ↔ {𝑋} <<s 𝑅))
17 fveqeq2 6852 . . . . . . 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 1437 . . . . . 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 7340 . . . 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 460 . . 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 581 . 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 397  w3a 1088   = wceq 1542  wcel 2107  ∃!wreu 3350  {crab 3406  {csn 4587   cint 4908   class class class wbr 5106  cima 5637  cfv 6497  crio 7313  (class class class)co 7358   No csur 27004   bday cbday 27006   <<s csslt 27142   |s cscut 27144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1o 8413  df-2o 8414  df-no 27007  df-slt 27008  df-bday 27009  df-sslt 27143  df-scut 27145
This theorem is referenced by:  eqscut2  27167  cuteq0  27193  madebdaylemlrcut  27250  cofcut1  27261
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