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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 6927). (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 14653 | . 2 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩)) | |
2 | 1 | mpondm0 7658 | 1 ⊢ (¬ (𝑆 ∈ V ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∅c0 4318 ⟨cop 4630 (class class class)co 7416 0cc0 11138 ℕ0cn0 12502 substr csubstr 14622 prefix cpfx 14652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-xp 5678 df-dm 5682 df-iota 6495 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-pfx 14653 |
This theorem is referenced by: pfxval0 14658 pfxnd0 14670 |
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