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.

Step	Hyp	Ref	Expression
1		2fveq3 6779	. . . 4 ⊢ (𝑦 = 𝐴 → ( O ‘( bday ‘𝑦)) = ( O ‘( bday ‘𝐴)))
2		breq2 5078	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝐴))
3	1, 2	rabeqbidv 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
6	5	rabex 5256	. . 3 ⊢ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} ∈ V
7	3, 4, 6	fvmpt 6875	. 2 ⊢ (𝐴 ∈ No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴})
8	4	fvmptndm 6905	. . 3 ⊢ (¬ 𝐴 ∈ No → ( L ‘𝐴) = ∅)
9		bdaydm 33969	. . . . . . . . 9 ⊢ dom bday = No
10	9	eleq2i 2830	. . . . . . . 8 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No )
11		ndmfv 6804	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ dom bday → ( bday ‘𝐴) = ∅)
12	10, 11	sylnbir 331	. . . . . . 7 ⊢ (¬ 𝐴 ∈ No → ( bday ‘𝐴) = ∅)
13	12	fveq2d 6778	. . . . . 6 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ( O ‘∅))
14		old0 34043	. . . . . 6 ⊢ ( O ‘∅) = ∅
15	13, 14	eqtrdi 2794	. . . . 5 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ∅)
16	15	rabeqdv 3419	. . . 4 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} = {𝑥 ∈ ∅ ∣ 𝑥 <s 𝐴})
17		rab0 4316	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝑥 <s 𝐴} = ∅
18	16, 17	eqtrdi 2794	. . 3 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} = ∅)
19	8, 18	eqtr4d 2781	. 2 ⊢ (¬ 𝐴 ∈ No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴})
20	7, 19	pm2.61i 182	1 ⊢ ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}

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