Theorem pfxnndmnd 14628

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 6920). (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 14627	. 2 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))
2	1	mpondm0 7644	1 ⊢ (¬ (𝑆 ∈ V ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  ∅c0 4317  ⟨cop 4629  (class class class)co 7405  0cc0 11112  ℕ₀cn0 12476   substr csubstr 14596   prefix cpfx 14626

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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-dm 5679  df-iota 6489  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-pfx 14627

This theorem is referenced by:  pfxval0  14632  pfxnd0  14644