Theorem pfxnndmnd 14826

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 6917). (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.

Step	Hyp	Ref	Expression
1		df-pfx 14825	. 2 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))
2	1	mpondm0 7661	1 ⊢ (¬ (𝑆 ∈ V ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ∅c0 4278  ⟨cop 4834  (class class class)co 7420  0cc0 11203  ℕ₀cn0 12609   substr csubstr 14792   prefix cpfx 14824

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-dm 5660  df-iota 6494  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-pfx 14825

This theorem is used by:  pfxval0  14830  pfxnd0  14842