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Theorem pfxnndmnd 14654
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.)
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 14653 . 2 prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))
21mpondm0 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  0cn0 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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