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Theorem lrold 27935
Description: The union of the left and right options of a surreal make its old set. (Contributed by Scott Fenton, 9-Oct-2024.)
Assertion
Ref Expression
lrold (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday 𝐴))

Proof of Theorem lrold
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 leftval 27902 . . . . 5 ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴}
2 rightval 27903 . . . . 5 ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥}
31, 2uneq12i 4166 . . . 4 (( L ‘𝐴) ∪ ( R ‘𝐴)) = ({𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥})
4 unrab 4315 . . . 4 ({𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥}) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ (𝑥 <s 𝐴𝐴 <s 𝑥)}
53, 4eqtri 2765 . . 3 (( L ‘𝐴) ∪ ( R ‘𝐴)) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ (𝑥 <s 𝐴𝐴 <s 𝑥)}
6 oldirr 27928 . . . . . . . 8 ¬ 𝐴 ∈ ( O ‘( bday 𝐴))
7 eleq1 2829 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ ( O ‘( bday 𝐴)) ↔ 𝐴 ∈ ( O ‘( bday 𝐴))))
86, 7mtbiri 327 . . . . . . 7 (𝑥 = 𝐴 → ¬ 𝑥 ∈ ( O ‘( bday 𝐴)))
98necon2ai 2970 . . . . . 6 (𝑥 ∈ ( O ‘( bday 𝐴)) → 𝑥𝐴)
109adantl 481 . . . . 5 ((𝐴 No 𝑥 ∈ ( O ‘( bday 𝐴))) → 𝑥𝐴)
11 oldssno 27900 . . . . . . 7 ( O ‘( bday 𝐴)) ⊆ No
1211sseli 3979 . . . . . 6 (𝑥 ∈ ( O ‘( bday 𝐴)) → 𝑥 No )
13 slttrine 27796 . . . . . . 7 ((𝑥 No 𝐴 No ) → (𝑥𝐴 ↔ (𝑥 <s 𝐴𝐴 <s 𝑥)))
1413ancoms 458 . . . . . 6 ((𝐴 No 𝑥 No ) → (𝑥𝐴 ↔ (𝑥 <s 𝐴𝐴 <s 𝑥)))
1512, 14sylan2 593 . . . . 5 ((𝐴 No 𝑥 ∈ ( O ‘( bday 𝐴))) → (𝑥𝐴 ↔ (𝑥 <s 𝐴𝐴 <s 𝑥)))
1610, 15mpbid 232 . . . 4 ((𝐴 No 𝑥 ∈ ( O ‘( bday 𝐴))) → (𝑥 <s 𝐴𝐴 <s 𝑥))
1716rabeqcda 3448 . . 3 (𝐴 No → {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ (𝑥 <s 𝐴𝐴 <s 𝑥)} = ( O ‘( bday 𝐴)))
185, 17eqtrid 2789 . 2 (𝐴 No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday 𝐴)))
19 un0 4394 . . 3 (∅ ∪ ∅) = ∅
20 leftf 27904 . . . . . . 7 L : No ⟶𝒫 No
2120fdmi 6747 . . . . . 6 dom L = No
2221eleq2i 2833 . . . . 5 (𝐴 ∈ dom L ↔ 𝐴 No )
23 ndmfv 6941 . . . . 5 𝐴 ∈ dom L → ( L ‘𝐴) = ∅)
2422, 23sylnbir 331 . . . 4 𝐴 No → ( L ‘𝐴) = ∅)
25 rightf 27905 . . . . . . 7 R : No ⟶𝒫 No
2625fdmi 6747 . . . . . 6 dom R = No
2726eleq2i 2833 . . . . 5 (𝐴 ∈ dom R ↔ 𝐴 No )
28 ndmfv 6941 . . . . 5 𝐴 ∈ dom R → ( R ‘𝐴) = ∅)
2927, 28sylnbir 331 . . . 4 𝐴 No → ( R ‘𝐴) = ∅)
3024, 29uneq12d 4169 . . 3 𝐴 No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = (∅ ∪ ∅))
31 bdaydm 27819 . . . . . . 7 dom bday = No
3231eleq2i 2833 . . . . . 6 (𝐴 ∈ dom bday 𝐴 No )
33 ndmfv 6941 . . . . . 6 𝐴 ∈ dom bday → ( bday 𝐴) = ∅)
3432, 33sylnbir 331 . . . . 5 𝐴 No → ( bday 𝐴) = ∅)
3534fveq2d 6910 . . . 4 𝐴 No → ( O ‘( bday 𝐴)) = ( O ‘∅))
36 old0 27898 . . . 4 ( O ‘∅) = ∅
3735, 36eqtrdi 2793 . . 3 𝐴 No → ( O ‘( bday 𝐴)) = ∅)
3819, 30, 373eqtr4a 2803 . 2 𝐴 No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday 𝐴)))
3918, 38pm2.61i 182 1 (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  {crab 3436  cun 3949  c0 4333  𝒫 cpw 4600   class class class wbr 5143  dom cdm 5685  cfv 6561   No csur 27684   <s cslt 27685   bday cbday 27686   O cold 27882   L cleft 27884   R cright 27885
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-made 27886  df-old 27887  df-left 27889  df-right 27890
This theorem is referenced by:  lruneq  27944  lrrecval2  27973  addsbdaylem  28049  negsbdaylem  28088  sltonold  28283
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