Theorem pfxnndmnd 14676

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 6888). (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 14675	. 2 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))
2	1	mpondm0 7625	1 ⊢ (¬ (𝑆 ∈ V ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1554   ∈ wcel 2136  Vcvv 3448  ∅c0 4280  ⟨cop 4582  (class class class)co 7385  0cc0 11063  ℕ₀cn0 12471   substr csubstr 14644   prefix cpfx 14674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-xp 5646  df-dm 5650  df-iota 6466  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-pfx 14675

This theorem is referenced by:  pfxval0  14680  pfxnd0  14692