Theorem disjwrdpfx 14744

Description: Sets of words are disjoint if each set contains exactly the extensions of distinct words of a fixed length. Remark: A word 𝑊 is called an "extension" of a word 𝑃 if 𝑃 is a prefix of 𝑊. (Contributed by AV, 29-Jul-2018.) (Revised by AV, 6-May-2020.)

Assertion

Ref	Expression
disjwrdpfx	⊢ Disj 𝑦 ∈ 𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}

Distinct variable groups:   𝑦,𝑁   𝑥,𝑉   𝑥,𝑦

Allowed substitution hints:   𝑁(𝑥)   𝑉(𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem disjwrdpfx

Step	Hyp	Ref	Expression
1		invdisjrab 5138	1 ⊢ Disj 𝑦 ∈ 𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}

Syntax hints:   = wceq 1539  {crab 3436  Disj wdisj 5118  (class class class)co 7438  Word cword 14558   prefix cpfx 14714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rmo 3380  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-disj 5119

This theorem is referenced by:  disjxwwlksn  29950  disjxwwlkn  29959