Theorem rightval 28241

Description: The value of the right options function. (Contributed by Scott Fenton, 9-Oct-2024.)

Assertion

Ref	Expression
rightval	⊢ (R‘𝐴) = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝐴 <s 𝑥}

Distinct variable group:   𝑥,𝐴

Proof of Theorem rightval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fveq3 6890	. . . 4 ⊢ (𝑦 = 𝐴 → (O‘(bday‘𝑦)) = (O‘(bday‘𝐴)))
2		breq1 5105	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 <s 𝑥 ↔ 𝐴 <s 𝑥))
3	1, 2	rabeqbidv 3429	. . 3 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ (O‘(bday‘𝑦)) ∣ 𝑦 <s 𝑥} = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝐴 <s 𝑥})
4		df-right 28222	. . 3 ⊢ R = (𝑦 ∈ No ↦ {𝑥 ∈ (O‘(bday‘𝑦)) ∣ 𝑦 <s 𝑥})
5		fvex 6898	. . . 4 ⊢ (O‘(bday‘𝐴)) ∈ V
6	5	rabex 5299	. . 3 ⊢ {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝐴 <s 𝑥} ∈ V
7	3, 4, 6	fvmpt 6993	. 2 ⊢ (𝐴 ∈ No → (R‘𝐴) = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝐴 <s 𝑥})
8	4	fvmptndm 7025	. . 3 ⊢ (¬ 𝐴 ∈ No → (R‘𝐴) = ∅)
9		bdaydm 28140	. . . . . . . . 9 ⊢ dom bday = No
10	9	eleq2i 2852	. . . . . . . 8 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No)
11		ndmfv 6917	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ dom bday → (bday‘𝐴) = ∅)
12	10, 11	sylnbir 334	. . . . . . 7 ⊢ (¬ 𝐴 ∈ No → (bday‘𝐴) = ∅)
13	12	fveq2d 6889	. . . . . 6 ⊢ (¬ 𝐴 ∈ No → (O‘(bday‘𝐴)) = (O‘∅))
14		old0 28230	. . . . . 6 ⊢ (O‘∅) = ∅
15	13, 14	eqtrdi 2811	. . . . 5 ⊢ (¬ 𝐴 ∈ No → (O‘(bday‘𝐴)) = ∅)
16	15	rabeqdv 3427	. . . 4 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝐴 <s 𝑥} = {𝑥 ∈ ∅ ∣ 𝐴 <s 𝑥})
17		rab0 4334	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝐴 <s 𝑥} = ∅
18	16, 17	eqtrdi 2811	. . 3 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝐴 <s 𝑥} = ∅)
19	8, 18	eqtr4d 2798	. 2 ⊢ (¬ 𝐴 ∈ No → (R‘𝐴) = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝐴 <s 𝑥})
20	7, 19	pm2.61i 184	1 ⊢ (R‘𝐴) = {𝑥 ∈ (O‘(bday‘𝐴)) ∣ 𝐴 <s 𝑥}

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2144  {crab 3412  ∅c0 4278   class class class wbr 5102  dom cdm 5650  ‘cfv 6538  Nocsur 28002   <s clts 28003  bdaycbday 28004  Ocold 28214  Rcright 28217

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-no 28005  df-bday 28007  df-made 28218  df-old 28219  df-right 28222

This theorem is used by:  elright  28243  sltsright  28252  rightssold  28263  right1s  28287  lrold  28288  madebdaylemlrcut  28290  cofcutr  28315  cofcutrtime  28318  addsproplem2  28361