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Mirrors > Home > MPE Home > Th. List > Mathboxes > rightval | Structured version Visualization version GIF version |
Description: The value of the right options function. (Contributed by Scott Fenton, 9-Oct-2024.) |
Ref | Expression |
---|---|
rightval | ⊢ ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2fveq3 6743 | . . . 4 ⊢ (𝑦 = 𝐴 → ( O ‘( bday ‘𝑦)) = ( O ‘( bday ‘𝐴))) | |
2 | breq1 5072 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 <s 𝑥 ↔ 𝐴 <s 𝑥)) | |
3 | 1, 2 | rabeqbidv 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 | |
6 | 5 | rabex 5241 | . . 3 ⊢ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥} ∈ V |
7 | 3, 4, 6 | fvmpt 6839 | . 2 ⊢ (𝐴 ∈ No → ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}) |
8 | 4 | fvmptndm 6869 | . . 3 ⊢ (¬ 𝐴 ∈ No → ( R ‘𝐴) = ∅) |
9 | bdaydm 33739 | . . . . . . . . 9 ⊢ dom bday = No | |
10 | 9 | eleq2i 2831 | . . . . . . . 8 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No ) |
11 | ndmfv 6768 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ dom bday → ( bday ‘𝐴) = ∅) | |
12 | 10, 11 | sylnbir 334 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ No → ( bday ‘𝐴) = ∅) |
13 | 12 | fveq2d 6742 | . . . . . 6 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ( O ‘∅)) |
14 | old0 33813 | . . . . . 6 ⊢ ( O ‘∅) = ∅ | |
15 | 13, 14 | eqtrdi 2796 | . . . . 5 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ∅) |
16 | 15 | rabeqdv 3410 | . . . 4 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥} = {𝑥 ∈ ∅ ∣ 𝐴 <s 𝑥}) |
17 | rab0 4313 | . . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝐴 <s 𝑥} = ∅ | |
18 | 16, 17 | eqtrdi 2796 | . . 3 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥} = ∅) |
19 | 8, 18 | eqtr4d 2782 | . 2 ⊢ (¬ 𝐴 ∈ No → ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}) |
20 | 7, 19 | pm2.61i 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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