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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 6907 | . . . 4 ⊢ (𝑦 = 𝐴 → ( O ‘( bday ‘𝑦)) = ( O ‘( bday ‘𝐴))) | |
2 | breq2 5156 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝐴)) | |
3 | 1, 2 | rabeqbidv 3448 | . . 3 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ ( O ‘( bday ‘𝑦)) ∣ 𝑥 <s 𝑦} = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}) |
4 | df-left 27797 | . . 3 ⊢ L = (𝑦 ∈ No ↦ {𝑥 ∈ ( O ‘( bday ‘𝑦)) ∣ 𝑥 <s 𝑦}) | |
5 | fvex 6915 | . . . 4 ⊢ ( O ‘( bday ‘𝐴)) ∈ V | |
6 | 5 | rabex 5338 | . . 3 ⊢ {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} ∈ V |
7 | 3, 4, 6 | fvmpt 7010 | . 2 ⊢ (𝐴 ∈ No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}) |
8 | 4 | fvmptndm 7041 | . . 3 ⊢ (¬ 𝐴 ∈ No → ( L ‘𝐴) = ∅) |
9 | bdaydm 27727 | . . . . . . . . 9 ⊢ dom bday = No | |
10 | 9 | eleq2i 2821 | . . . . . . . 8 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No ) |
11 | ndmfv 6937 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ dom bday → ( bday ‘𝐴) = ∅) | |
12 | 10, 11 | sylnbir 330 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ No → ( bday ‘𝐴) = ∅) |
13 | 12 | fveq2d 6906 | . . . . . 6 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ( O ‘∅)) |
14 | old0 27806 | . . . . . 6 ⊢ ( O ‘∅) = ∅ | |
15 | 13, 14 | eqtrdi 2784 | . . . . 5 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ∅) |
16 | 15 | rabeqdv 3446 | . . . 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 2784 | . . 3 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} = ∅) |
19 | 8, 18 | eqtr4d 2771 | . 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 1533 ∈ wcel 2098 {crab 3430 ∅c0 4326 class class class wbr 5152 dom cdm 5682 ‘cfv 6553 No csur 27593 <s cslt 27594 bday cbday 27595 O cold 27790 L cleft 27792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-1o 8493 df-no 27596 df-bday 27598 df-made 27794 df-old 27795 df-left 27797 |
This theorem is referenced by: ssltleft 27817 leftssold 27825 left1s 27841 lrold 27843 madebdaylemlrcut 27845 sltlpss 27853 0elleft 27856 cofcutr 27864 cofcutrtime 27867 addsproplem2 27907 addsproplem4 27909 addsproplem6 27911 sleadd1 27926 negsproplem4 27963 negsproplem6 27965 negsid 27973 mulsproplem12 28047 precsexlem9 28133 sltonold 28173 |
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