Theorem leftval 28000

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 6876	. . . 4 ⊢ (𝑦 = 𝐴 → ( O ‘( bday ‘𝑦)) = ( O ‘( bday ‘𝐴)))
2		breq2 5109	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝐴))
3	1, 2	rabeqbidv 3435	. . 3 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ ( O ‘( bday ‘𝑦)) ∣ 𝑥 <s 𝑦} = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴})
4		df-left 27981	. . 3 ⊢ L = (𝑦 ∈ No ↦ {𝑥 ∈ ( O ‘( bday ‘𝑦)) ∣ 𝑥 <s 𝑦})
5		fvex 6884	. . . 4 ⊢ ( O ‘( bday ‘𝐴)) ∈ V
6	5	rabex 5300	. . 3 ⊢ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} ∈ V
7	3, 4, 6	fvmpt 6979	. 2 ⊢ (𝐴 ∈ No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴})
8	4	fvmptndm 7011	. . 3 ⊢ (¬ 𝐴 ∈ No → ( L ‘𝐴) = ∅)
9		bdaydm 27900	. . . . . . . . 9 ⊢ dom bday = No
10	9	eleq2i 2857	. . . . . . . 8 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No )
11		ndmfv 6903	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ dom bday → ( bday ‘𝐴) = ∅)
12	10, 11	sylnbir 334	. . . . . . 7 ⊢ (¬ 𝐴 ∈ No → ( bday ‘𝐴) = ∅)
13	12	fveq2d 6875	. . . . . 6 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ( O ‘∅))
14		old0 27990	. . . . . 6 ⊢ ( O ‘∅) = ∅
15	13, 14	eqtrdi 2816	. . . . 5 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ∅)
16	15	rabeqdv 3432	. . . 4 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} = {𝑥 ∈ ∅ ∣ 𝑥 <s 𝐴})
17		rab0 4342	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝑥 <s 𝐴} = ∅
18	16, 17	eqtrdi 2816	. . 3 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} = ∅)
19	8, 18	eqtr4d 2803	. 2 ⊢ (¬ 𝐴 ∈ No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴})
20	7, 19	pm2.61i 184	1 ⊢ ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}

Syntax hints:  ¬ wn 3   = wceq 1563   ∈ wcel 2145  {crab 3417  ∅c0 4288   class class class wbr 5105  dom cdm 5652  ‘cfv 6525   No csur 27762   <s clts 27763   bday cbday 27764   O cold 27974   L cleft 27976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-1o 8441  df-no 27765  df-bday 27767  df-made 27978  df-old 27979  df-left 27981

This theorem is referenced by:  elleft  28002  sltsleft  28011  leftssold  28022  left1s  28046  lrold  28048  madebdaylemlrcut  28050  ltslpss  28059  cofcutr  28075  cofcutrtime  28078  addsproplem2  28121