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Mirrors > Home > MPE Home > Th. List > pfx0 | Structured version Visualization version GIF version |
Description: A prefix of an empty set is always the empty set. (Contributed by AV, 3-May-2020.) |
Ref | Expression |
---|---|
pfx0 | ⊢ (∅ prefix 𝐿) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 5586 | . . . 4 ⊢ (〈∅, 𝐿〉 ∈ (V × ℕ0) ↔ (∅ ∈ V ∧ 𝐿 ∈ ℕ0)) | |
2 | pfxval 14029 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝐿 ∈ ℕ0) → (∅ prefix 𝐿) = (∅ substr 〈0, 𝐿〉)) | |
3 | swrd0 14014 | . . . . 5 ⊢ (∅ substr 〈0, 𝐿〉) = ∅ | |
4 | 2, 3 | syl6eq 2872 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐿 ∈ ℕ0) → (∅ prefix 𝐿) = ∅) |
5 | 1, 4 | sylbi 219 | . . 3 ⊢ (〈∅, 𝐿〉 ∈ (V × ℕ0) → (∅ prefix 𝐿) = ∅) |
6 | df-pfx 14027 | . . . 4 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr 〈0, 𝑙〉)) | |
7 | ovex 7183 | . . . 4 ⊢ (𝑠 substr 〈0, 𝑙〉) ∈ V | |
8 | 6, 7 | dmmpo 7763 | . . 3 ⊢ dom prefix = (V × ℕ0) |
9 | 5, 8 | eleq2s 2931 | . 2 ⊢ (〈∅, 𝐿〉 ∈ dom prefix → (∅ prefix 𝐿) = ∅) |
10 | df-ov 7153 | . . 3 ⊢ (∅ prefix 𝐿) = ( prefix ‘〈∅, 𝐿〉) | |
11 | ndmfv 6695 | . . 3 ⊢ (¬ 〈∅, 𝐿〉 ∈ dom prefix → ( prefix ‘〈∅, 𝐿〉) = ∅) | |
12 | 10, 11 | syl5eq 2868 | . 2 ⊢ (¬ 〈∅, 𝐿〉 ∈ dom prefix → (∅ prefix 𝐿) = ∅) |
13 | 9, 12 | pm2.61i 184 | 1 ⊢ (∅ prefix 𝐿) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∅c0 4291 〈cop 4567 × cxp 5548 dom cdm 5550 ‘cfv 6350 (class class class)co 7150 0cc0 10531 ℕ0cn0 11891 substr csubstr 13996 prefix cpfx 14026 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-substr 13997 df-pfx 14027 |
This theorem is referenced by: cshword 14147 |
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