Theorem pfxnndmnd 14775

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 6906). (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 14774	. 2 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))
2	1	mpondm0 7650	1 ⊢ (¬ (𝑆 ∈ V ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ∅c0 4279  ⟨cop 4590  (class class class)co 7409  0cc0 11157  ℕ₀cn0 12561   substr csubstr 14741   prefix cpfx 14773

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5654  df-dm 5658  df-iota 6484  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-pfx 14774

This theorem is used by:  pfxval0  14779  pfxnd0  14791