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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 6913). (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 14603 | . 2 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr 〈0, 𝑙〉)) | |
2 | 1 | mpondm0 7630 | 1 ⊢ (¬ (𝑆 ∈ V ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3473 ∅c0 4318 〈cop 4628 (class class class)co 7393 0cc0 11092 ℕ0cn0 12454 substr csubstr 14572 prefix cpfx 14602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-xp 5675 df-dm 5679 df-iota 6484 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-pfx 14603 |
This theorem is referenced by: pfxval0 14608 pfxnd0 14620 |
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