Theorem leftval 28268

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 6916	. . . 4 ⊢ (𝑦 = 𝐴 → (O‘(bday‘𝑦)) = (O‘(bday‘𝐴)))
2		breq2 5110	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝐴))
3	1, 2	rabeqbidv 3433	. . 3 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ (O‘(bday‘𝑦)) ∣ 𝑥 <s 𝑦} = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝑥 <s 𝐴})
4		df-left 28249	. . 3 ⊢ L = (𝑦 ∈ No ↦ {𝑥 ∈ (O‘(bday‘𝑦)) ∣ 𝑥 <s 𝑦})
5		fvex 6924	. . . 4 ⊢ (O‘(bday‘𝐴)) ∈ V
6	5	rabex 5305	. . 3 ⊢ {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝑥 <s 𝐴} ∈ V
7	3, 4, 6	fvmpt 7019	. 2 ⊢ (𝐴 ∈ No → (L‘𝐴) = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝑥 <s 𝐴})
8	4	fvmptndm 7051	. . 3 ⊢ (¬ 𝐴 ∈ No → (L‘𝐴) = ∅)
9		bdaydm 28168	. . . . . . . . 9 ⊢ dom bday = No
10	9	eleq2i 2856	. . . . . . . 8 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No)
11		ndmfv 6943	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ dom bday → (bday‘𝐴) = ∅)
12	10, 11	sylnbir 334	. . . . . . 7 ⊢ (¬ 𝐴 ∈ No → (bday‘𝐴) = ∅)
13	12	fveq2d 6915	. . . . . 6 ⊢ (¬ 𝐴 ∈ No → (O‘(bday‘𝐴)) = (O‘∅))
14		old0 28258	. . . . . 6 ⊢ (O‘∅) = ∅
15	13, 14	eqtrdi 2814	. . . . 5 ⊢ (¬ 𝐴 ∈ No → (O‘(bday‘𝐴)) = ∅)
16	15	rabeqdv 3431	. . . 4 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝑥 <s 𝐴} = {𝑥 ∈ ∅ ∣ 𝑥 <s 𝐴})
17		rab0 4338	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝑥 <s 𝐴} = ∅
18	16, 17	eqtrdi 2814	. . 3 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝑥 <s 𝐴} = ∅)
19	8, 18	eqtr4d 2801	. 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}

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147  {crab 3416  ∅c0 4281   class class class wbr 5106  dom cdm 5656  ‘cfv 6561  Nocsur 28030   <s clts 28031  bdaycbday 28032  Ocold 28242  Lcleft 28244

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-no 28033  df-bday 28035  df-made 28246  df-old 28247  df-left 28249

This theorem is used by:  elleft  28270  sltsleft  28279  leftssold  28290  left1s  28314  lrold  28316  madebdaylemlrcut  28318  ltslpss  28327  cofcutr  28343  cofcutrtime  28346  addsproplem2  28389