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Theorem rightval 33818
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.
StepHypRef Expression
1 2fveq3 6743 . . . 4 (𝑦 = 𝐴 → ( O ‘( bday 𝑦)) = ( O ‘( bday 𝐴)))
2 breq1 5072 . . . 4 (𝑦 = 𝐴 → (𝑦 <s 𝑥𝐴 <s 𝑥))
31, 2rabeqbidv 3411 . . 3 (𝑦 = 𝐴 → {𝑥 ∈ ( O ‘( bday 𝑦)) ∣ 𝑦 <s 𝑥} = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥})
4 df-right 33805 . . 3 R = (𝑦 No ↦ {𝑥 ∈ ( O ‘( bday 𝑦)) ∣ 𝑦 <s 𝑥})
5 fvex 6751 . . . 4 ( O ‘( bday 𝐴)) ∈ V
65rabex 5241 . . 3 {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥} ∈ V
73, 4, 6fvmpt 6839 . 2 (𝐴 No → ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥})
84fvmptndm 6869 . . 3 𝐴 No → ( R ‘𝐴) = ∅)
9 bdaydm 33739 . . . . . . . . 9 dom bday = No
109eleq2i 2831 . . . . . . . 8 (𝐴 ∈ dom bday 𝐴 No )
11 ndmfv 6768 . . . . . . . 8 𝐴 ∈ dom bday → ( bday 𝐴) = ∅)
1210, 11sylnbir 334 . . . . . . 7 𝐴 No → ( bday 𝐴) = ∅)
1312fveq2d 6742 . . . . . 6 𝐴 No → ( O ‘( bday 𝐴)) = ( O ‘∅))
14 old0 33813 . . . . . 6 ( O ‘∅) = ∅
1513, 14eqtrdi 2796 . . . . 5 𝐴 No → ( O ‘( bday 𝐴)) = ∅)
1615rabeqdv 3410 . . . 4 𝐴 No → {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥} = {𝑥 ∈ ∅ ∣ 𝐴 <s 𝑥})
17 rab0 4313 . . . 4 {𝑥 ∈ ∅ ∣ 𝐴 <s 𝑥} = ∅
1816, 17eqtrdi 2796 . . 3 𝐴 No → {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥} = ∅)
198, 18eqtr4d 2782 . 2 𝐴 No → ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥})
207, 19pm2.61i 185 1 ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1543  wcel 2112  {crab 3068  c0 4253   class class class wbr 5069  dom cdm 5568  cfv 6400   No csur 33613   <s cslt 33614   bday cbday 33615   O cold 33797   R cright 33800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5195  ax-sep 5208  ax-nul 5215  ax-pr 5338  ax-un 7544
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4836  df-iun 4922  df-br 5070  df-opab 5132  df-mpt 5152  df-tr 5178  df-id 5471  df-eprel 5477  df-po 5485  df-so 5486  df-fr 5526  df-we 5528  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-pred 6178  df-ord 6236  df-on 6237  df-suc 6239  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408  df-wrecs 8070  df-recs 8131  df-1o 8225  df-no 33616  df-bday 33618  df-made 33801  df-old 33802  df-right 33805
This theorem is referenced by:  ssltright  33825  rightssold  33832  lrold  33847  madebdaylemlrcut  33849  cofcutr  33862  cofcutrtime  33863
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