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Theorem pfxnndmnd 14604
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.)
Assertion
Ref Expression
pfxnndmnd (¬ (𝑆 ∈ V ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = ∅)

Proof of Theorem pfxnndmnd
Dummy variables 𝑠 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pfx 14603 . 2 prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))
21mpondm0 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  0cn0 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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