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Theorem leftval 34047
Description: The value of the left options function. (Contributed by Scott Fenton, 9-Oct-2024.)
Assertion
Ref Expression
leftval ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴}
Distinct variable group:   𝑥,𝐴

Proof of Theorem leftval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 2fveq3 6779 . . . 4 (𝑦 = 𝐴 → ( O ‘( bday 𝑦)) = ( O ‘( bday 𝐴)))
2 breq2 5078 . . . 4 (𝑦 = 𝐴 → (𝑥 <s 𝑦𝑥 <s 𝐴))
31, 2rabeqbidv 3420 . . 3 (𝑦 = 𝐴 → {𝑥 ∈ ( O ‘( bday 𝑦)) ∣ 𝑥 <s 𝑦} = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴})
4 df-left 34034 . . 3 L = (𝑦 No ↦ {𝑥 ∈ ( O ‘( bday 𝑦)) ∣ 𝑥 <s 𝑦})
5 fvex 6787 . . . 4 ( O ‘( bday 𝐴)) ∈ V
65rabex 5256 . . 3 {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴} ∈ V
73, 4, 6fvmpt 6875 . 2 (𝐴 No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴})
84fvmptndm 6905 . . 3 𝐴 No → ( L ‘𝐴) = ∅)
9 bdaydm 33969 . . . . . . . . 9 dom bday = No
109eleq2i 2830 . . . . . . . 8 (𝐴 ∈ dom bday 𝐴 No )
11 ndmfv 6804 . . . . . . . 8 𝐴 ∈ dom bday → ( bday 𝐴) = ∅)
1210, 11sylnbir 331 . . . . . . 7 𝐴 No → ( bday 𝐴) = ∅)
1312fveq2d 6778 . . . . . 6 𝐴 No → ( O ‘( bday 𝐴)) = ( O ‘∅))
14 old0 34043 . . . . . 6 ( O ‘∅) = ∅
1513, 14eqtrdi 2794 . . . . 5 𝐴 No → ( O ‘( bday 𝐴)) = ∅)
1615rabeqdv 3419 . . . 4 𝐴 No → {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴} = {𝑥 ∈ ∅ ∣ 𝑥 <s 𝐴})
17 rab0 4316 . . . 4 {𝑥 ∈ ∅ ∣ 𝑥 <s 𝐴} = ∅
1816, 17eqtrdi 2794 . . 3 𝐴 No → {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴} = ∅)
198, 18eqtr4d 2781 . 2 𝐴 No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴})
207, 19pm2.61i 182 1 ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1539  wcel 2106  {crab 3068  c0 4256   class class class wbr 5074  dom cdm 5589  cfv 6433   No csur 33843   <s cslt 33844   bday cbday 33845   O cold 34027   L cleft 34029
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-1o 8297  df-no 33846  df-bday 33848  df-made 34031  df-old 34032  df-left 34034
This theorem is referenced by:  ssltleft  34054  leftssold  34061  lrold  34077  madebdaylemlrcut  34079  sltlpss  34087  cofcutr  34092  cofcutrtime  34093
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