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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 6912 | . . . 4 ⊢ (𝑦 = 𝐴 → ( O ‘( bday ‘𝑦)) = ( O ‘( bday ‘𝐴))) | |
2 | breq2 5152 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝐴)) | |
3 | 1, 2 | rabeqbidv 3452 | . . 3 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ ( O ‘( bday ‘𝑦)) ∣ 𝑥 <s 𝑦} = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}) |
4 | df-left 27904 | . . 3 ⊢ L = (𝑦 ∈ No ↦ {𝑥 ∈ ( O ‘( bday ‘𝑦)) ∣ 𝑥 <s 𝑦}) | |
5 | fvex 6920 | . . . 4 ⊢ ( O ‘( bday ‘𝐴)) ∈ V | |
6 | 5 | rabex 5345 | . . 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 27834 | . . . . . . . . 9 ⊢ dom bday = No | |
10 | 9 | eleq2i 2831 | . . . . . . . 8 ⊢ (𝐴 ∈ dom bday ↔ 𝐴 ∈ No ) |
11 | ndmfv 6942 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ dom bday → ( bday ‘𝐴) = ∅) | |
12 | 10, 11 | sylnbir 331 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ No → ( bday ‘𝐴) = ∅) |
13 | 12 | fveq2d 6911 | . . . . . 6 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ( O ‘∅)) |
14 | old0 27913 | . . . . . 6 ⊢ ( O ‘∅) = ∅ | |
15 | 13, 14 | eqtrdi 2791 | . . . . 5 ⊢ (¬ 𝐴 ∈ No → ( O ‘( bday ‘𝐴)) = ∅) |
16 | 15 | rabeqdv 3449 | . . . 4 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} = {𝑥 ∈ ∅ ∣ 𝑥 <s 𝐴}) |
17 | rab0 4392 | . . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝑥 <s 𝐴} = ∅ | |
18 | 16, 17 | eqtrdi 2791 | . . 3 ⊢ (¬ 𝐴 ∈ No → {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴} = ∅) |
19 | 8, 18 | eqtr4d 2778 | . 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 1537 ∈ wcel 2106 {crab 3433 ∅c0 4339 class class class wbr 5148 dom cdm 5689 ‘cfv 6563 No csur 27699 <s cslt 27700 bday cbday 27701 O cold 27897 L cleft 27899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-1o 8505 df-no 27702 df-bday 27704 df-made 27901 df-old 27902 df-left 27904 |
This theorem is referenced by: ssltleft 27924 leftssold 27932 left1s 27948 lrold 27950 madebdaylemlrcut 27952 sltlpss 27960 0elleft 27963 cofcutr 27973 cofcutrtime 27976 addsproplem2 28018 addsproplem4 28020 addsproplem6 28022 sleadd1 28037 negsproplem4 28078 negsproplem6 28080 negsid 28088 mulsproplem12 28168 precsexlem9 28254 sltonold 28298 |
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