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Mirrors > Home > MPE Home > Th. List > pfxnndmnd | Structured version Visualization version GIF version |
Description: The value of a prefix operation for out-of-domain arguments. (This is due to our definition of function values for out-of-domain arguments, see ndmfv 6693). (Contributed by AV, 3-Dec-2022.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pfxnndmnd | ⊢ (¬ (𝑆 ∈ V ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pfx 14028 | . 2 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr 〈0, 𝑙〉)) | |
2 | 1 | mpondm0 7379 | 1 ⊢ (¬ (𝑆 ∈ V ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∅c0 4284 〈cop 4566 (class class class)co 7149 0cc0 10530 ℕ0cn0 11891 substr csubstr 13997 prefix cpfx 14027 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-xp 5554 df-dm 5558 df-iota 6307 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-pfx 14028 |
This theorem is referenced by: pfxval0 14033 pfxnd0 14045 |
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