Theorem pfxnndmnd 14658

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 6931). (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 14657	. 2 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))
2	1	mpondm0 7661	1 ⊢ (¬ (𝑆 ∈ V ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3461  ∅c0 4322  ⟨cop 4636  (class class class)co 7419  0cc0 11140  ℕ₀cn0 12505   substr csubstr 14626   prefix cpfx 14656

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 5300  ax-nul 5307  ax-pr 5429

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 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-dm 5688  df-iota 6501  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-pfx 14657

This theorem is referenced by:  pfxval0  14662  pfxnd0  14674

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