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| Mirrors > Home > MPE Home > Th. List > rightf | Structured version Visualization version GIF version | ||
| Description: The functionality of the right options function. (Contributed by Scott Fenton, 6-Aug-2024.) |
| Ref | Expression |
|---|---|
| rightf | ⊢ R : No ⟶𝒫 No |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-right 27890 | . 2 ⊢ R = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦}) | |
| 2 | bdayelon 27821 | . . . . . . . 8 ⊢ ( bday ‘𝑥) ∈ On | |
| 3 | oldf 27896 | . . . . . . . . 9 ⊢ O :On⟶𝒫 No | |
| 4 | 3 | ffvelcdmi 7103 | . . . . . . . 8 ⊢ (( bday ‘𝑥) ∈ On → ( O ‘( bday ‘𝑥)) ∈ 𝒫 No ) |
| 5 | 2, 4 | mp1i 13 | . . . . . . 7 ⊢ (𝑥 ∈ No → ( O ‘( bday ‘𝑥)) ∈ 𝒫 No ) |
| 6 | 5 | elpwid 4609 | . . . . . 6 ⊢ (𝑥 ∈ No → ( O ‘( bday ‘𝑥)) ⊆ No ) |
| 7 | 6 | sselda 3983 | . . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ ( O ‘( bday ‘𝑥))) → 𝑦 ∈ No ) |
| 8 | 7 | a1d 25 | . . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ ( O ‘( bday ‘𝑥))) → (𝑥 <s 𝑦 → 𝑦 ∈ No )) |
| 9 | 8 | ralrimiva 3146 | . . 3 ⊢ (𝑥 ∈ No → ∀𝑦 ∈ ( O ‘( bday ‘𝑥))(𝑥 <s 𝑦 → 𝑦 ∈ No )) |
| 10 | fvex 6919 | . . . . . 6 ⊢ ( O ‘( bday ‘𝑥)) ∈ V | |
| 11 | 10 | rabex 5339 | . . . . 5 ⊢ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦} ∈ V |
| 12 | 11 | elpw 4604 | . . . 4 ⊢ ({𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦} ∈ 𝒫 No ↔ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦} ⊆ No ) |
| 13 | rabss 4072 | . . . 4 ⊢ ({𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦} ⊆ No ↔ ∀𝑦 ∈ ( O ‘( bday ‘𝑥))(𝑥 <s 𝑦 → 𝑦 ∈ No )) | |
| 14 | 12, 13 | bitri 275 | . . 3 ⊢ ({𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦} ∈ 𝒫 No ↔ ∀𝑦 ∈ ( O ‘( bday ‘𝑥))(𝑥 <s 𝑦 → 𝑦 ∈ No )) |
| 15 | 9, 14 | sylibr 234 | . 2 ⊢ (𝑥 ∈ No → {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦} ∈ 𝒫 No ) |
| 16 | 1, 15 | fmpti 7132 | 1 ⊢ R : No ⟶𝒫 No |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⊆ wss 3951 𝒫 cpw 4600 class class class wbr 5143 Oncon0 6384 ⟶wf 6557 ‘cfv 6561 No csur 27684 <s cslt 27685 bday cbday 27686 O cold 27882 R cright 27885 |
| 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-rmo 3380 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-tp 4631 df-op 4633 df-uni 4908 df-int 4947 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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-2o 8507 df-no 27687 df-slt 27688 df-bday 27689 df-sslt 27826 df-scut 27828 df-made 27886 df-old 27887 df-right 27890 |
| This theorem is referenced by: ssltright 27910 lltropt 27911 lrold 27935 |
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