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Mirrors > Home > MPE Home > Th. List > Mathboxes > leftf | Structured version Visualization version GIF version |
Description: The functionality of the left options function. (Contributed by Scott Fenton, 6-Aug-2024.) |
Ref | Expression |
---|---|
leftf | ⊢ L : No ⟶𝒫 No |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-left 34013 | . 2 ⊢ L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥}) | |
2 | bdayelon 33950 | . . . . . . . 8 ⊢ ( bday ‘𝑥) ∈ On | |
3 | oldf 34020 | . . . . . . . . 9 ⊢ O :On⟶𝒫 No | |
4 | 3 | ffvelrni 6954 | . . . . . . . 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 4549 | . . . . . 6 ⊢ (𝑥 ∈ No → ( O ‘( bday ‘𝑥)) ⊆ No ) |
7 | 6 | sselda 3925 | . . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ ( O ‘( bday ‘𝑥))) → 𝑦 ∈ No ) |
8 | 7 | a1d 25 | . . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ ( O ‘( bday ‘𝑥))) → (𝑦 <s 𝑥 → 𝑦 ∈ No )) |
9 | 8 | ralrimiva 3109 | . . 3 ⊢ (𝑥 ∈ No → ∀𝑦 ∈ ( O ‘( bday ‘𝑥))(𝑦 <s 𝑥 → 𝑦 ∈ No )) |
10 | fvex 6781 | . . . . . 6 ⊢ ( O ‘( bday ‘𝑥)) ∈ V | |
11 | 10 | rabex 5259 | . . . . 5 ⊢ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥} ∈ V |
12 | 11 | elpw 4542 | . . . 4 ⊢ ({𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥} ∈ 𝒫 No ↔ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥} ⊆ No ) |
13 | rabss 4009 | . . . 4 ⊢ ({𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥} ⊆ No ↔ ∀𝑦 ∈ ( O ‘( bday ‘𝑥))(𝑦 <s 𝑥 → 𝑦 ∈ No )) | |
14 | 12, 13 | bitri 274 | . . 3 ⊢ ({𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥} ∈ 𝒫 No ↔ ∀𝑦 ∈ ( O ‘( bday ‘𝑥))(𝑦 <s 𝑥 → 𝑦 ∈ No )) |
15 | 9, 14 | sylibr 233 | . 2 ⊢ (𝑥 ∈ No → {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥} ∈ 𝒫 No ) |
16 | 1, 15 | fmpti 6980 | 1 ⊢ L : No ⟶𝒫 No |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2109 ∀wral 3065 {crab 3069 ⊆ wss 3891 𝒫 cpw 4538 class class class wbr 5078 Oncon0 6263 ⟶wf 6426 ‘cfv 6430 No csur 33822 <s cslt 33823 bday cbday 33824 O cold 34006 L cleft 34008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-1o 8281 df-2o 8282 df-no 33825 df-slt 33826 df-bday 33827 df-sslt 33955 df-scut 33957 df-made 34010 df-old 34011 df-left 34013 |
This theorem is referenced by: ssltleft 34033 lrold 34056 |
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