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| Mirrors > Home > MPE Home > Th. List > leftval | Structured version Visualization version GIF version | ||
| Description: The value of the left options function. (Contributed by Scott Fenton, 9-Oct-2024.) |
| Ref | Expression |
|---|---|
| leftval | ⊢ ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2fveq3 6911 | . . . 4 ⊢ (𝑦 = 𝐴 → ( O ‘( bday ‘𝑦)) = ( O ‘( bday ‘𝐴))) | |
| 2 | breq2 5147 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝐴)) | |
| 3 | 1, 2 | rabeqbidv 3455 | . . 3 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ ( O ‘( bday ‘𝑦)) ∣ 𝑥 <s 𝑦} = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}) |
| 4 | df-left 27889 | . . 3 ⊢ L = (𝑦 ∈ No ↦ {𝑥 ∈ ( O ‘( bday ‘𝑦)) ∣ 𝑥 <s 𝑦}) | |
| 5 | fvex 6919 | . . . 4 ⊢ ( O ‘( bday ‘𝐴)) ∈ V | |
| 6 | 5 | rabex 5339 | . . 3 ⊢ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} ∈ V |
| 7 | 3, 4, 6 | fvmpt 7016 | . 2 ⊢ (𝐴 ∈ No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}) |
| 8 | 4 | fvmptndm 7047 | . . 3 ⊢ (¬ 𝐴 ∈ No → ( L ‘𝐴) = ∅) |
| 9 | bdaydm 27819 | . . . . . . . . 9 ⊢ dom bday = No | |
| 10 | 9 | eleq2i 2833 | . . . . . . . 8 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No ) |
| 11 | ndmfv 6941 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ dom bday → ( bday ‘𝐴) = ∅) | |
| 12 | 10, 11 | sylnbir 331 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ No → ( bday ‘𝐴) = ∅) |
| 13 | 12 | fveq2d 6910 | . . . . . 6 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ( O ‘∅)) |
| 14 | old0 27898 | . . . . . 6 ⊢ ( O ‘∅) = ∅ | |
| 15 | 13, 14 | eqtrdi 2793 | . . . . 5 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ∅) |
| 16 | 15 | rabeqdv 3452 | . . . 4 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} = {𝑥 ∈ ∅ ∣ 𝑥 <s 𝐴}) |
| 17 | rab0 4386 | . . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝑥 <s 𝐴} = ∅ | |
| 18 | 16, 17 | eqtrdi 2793 | . . 3 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} = ∅) |
| 19 | 8, 18 | eqtr4d 2780 | . 2 ⊢ (¬ 𝐴 ∈ No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}) |
| 20 | 7, 19 | pm2.61i 182 | 1 ⊢ ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 {crab 3436 ∅c0 4333 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 No csur 27684 <s cslt 27685 bday cbday 27686 O cold 27882 L cleft 27884 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-no 27687 df-bday 27689 df-made 27886 df-old 27887 df-left 27889 |
| This theorem is referenced by: ssltleft 27909 leftssold 27917 left1s 27933 lrold 27935 madebdaylemlrcut 27937 sltlpss 27945 0elleft 27948 cofcutr 27958 cofcutrtime 27961 addsproplem2 28003 addsproplem4 28005 addsproplem6 28007 sleadd1 28022 negsproplem4 28063 negsproplem6 28065 negsid 28073 mulsproplem12 28153 precsexlem9 28239 sltonold 28283 |
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