Theorem disjwrdpfx 14733

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 5096	1 ⊢ Disj 𝑦 ∈ 𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}

Syntax hints:   = wceq 1570  {crab 3416  Disj wdisj 5076  (class class class)co 7410  Word cword 14546   prefix cpfx 14704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rmo 3369  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-disj 5077

This theorem is referenced by:  disjxwwlksn  30253  disjxwwlkn  30262