Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pfx00 | Structured version Visualization version GIF version |
Description: The zero length prefix is the empty set. (Contributed by AV, 2-May-2020.) |
Ref | Expression |
---|---|
pfx00 | ⊢ (𝑆 prefix 0) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 5584 | . . . 4 ⊢ (〈𝑆, 0〉 ∈ (V × ℕ0) ↔ (𝑆 ∈ V ∧ 0 ∈ ℕ0)) | |
2 | pfxval 14027 | . . . . 5 ⊢ ((𝑆 ∈ V ∧ 0 ∈ ℕ0) → (𝑆 prefix 0) = (𝑆 substr 〈0, 0〉)) | |
3 | swrd00 13998 | . . . . 5 ⊢ (𝑆 substr 〈0, 0〉) = ∅ | |
4 | 2, 3 | syl6eq 2870 | . . . 4 ⊢ ((𝑆 ∈ V ∧ 0 ∈ ℕ0) → (𝑆 prefix 0) = ∅) |
5 | 1, 4 | sylbi 219 | . . 3 ⊢ (〈𝑆, 0〉 ∈ (V × ℕ0) → (𝑆 prefix 0) = ∅) |
6 | df-pfx 14025 | . . . 4 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr 〈0, 𝑙〉)) | |
7 | ovex 7181 | . . . 4 ⊢ (𝑠 substr 〈0, 𝑙〉) ∈ V | |
8 | 6, 7 | dmmpo 7761 | . . 3 ⊢ dom prefix = (V × ℕ0) |
9 | 5, 8 | eleq2s 2929 | . 2 ⊢ (〈𝑆, 0〉 ∈ dom prefix → (𝑆 prefix 0) = ∅) |
10 | df-ov 7151 | . . 3 ⊢ (𝑆 prefix 0) = ( prefix ‘〈𝑆, 0〉) | |
11 | ndmfv 6693 | . . 3 ⊢ (¬ 〈𝑆, 0〉 ∈ dom prefix → ( prefix ‘〈𝑆, 0〉) = ∅) | |
12 | 10, 11 | syl5eq 2866 | . 2 ⊢ (¬ 〈𝑆, 0〉 ∈ dom prefix → (𝑆 prefix 0) = ∅) |
13 | 9, 12 | pm2.61i 184 | 1 ⊢ (𝑆 prefix 0) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1531 ∈ wcel 2108 Vcvv 3493 ∅c0 4289 〈cop 4565 × cxp 5546 dom cdm 5548 ‘cfv 6348 (class class class)co 7148 0cc0 10529 ℕ0cn0 11889 substr csubstr 13994 prefix cpfx 14024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-fzo 13026 df-substr 13995 df-pfx 14025 |
This theorem is referenced by: cshw0 14148 |
Copyright terms: Public domain | W3C validator |