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Theorem reuccatpfxs1 41759
Description: There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. Could replace reuccats1 13526. (Contributed by AV, 10-May-2020.)
Assertion
Ref Expression
reuccatpfxs1 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑤𝑋 𝑊 = (𝑤 prefix (#‘𝑊))))
Distinct variable groups:   𝑣,𝑉,𝑤,𝑥   𝑣,𝑊,𝑤,𝑥   𝑣,𝑋,𝑤,𝑥

Proof of Theorem reuccatpfxs1
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 s1eq 13416 . . . . 5 (𝑣 = 𝑢 → ⟨“𝑣”⟩ = ⟨“𝑢”⟩)
21oveq2d 6706 . . . 4 (𝑣 = 𝑢 → (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“𝑢”⟩))
32eleq1d 2715 . . 3 (𝑣 = 𝑢 → ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ (𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋))
43reu8 3435 . 2 (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ ∃𝑣𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢)))
5 simprl 809 . . . . 5 ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) → (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋)
6 simpl 472 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → 𝑊 ∈ Word 𝑉)
76ad2antrr 762 . . . . . . . . 9 ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) → 𝑊 ∈ Word 𝑉)
87anim1i 591 . . . . . . . 8 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) → (𝑊 ∈ Word 𝑉𝑤𝑋))
9 simplrr 818 . . . . . . . 8 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) → ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))
10 simp-4r 824 . . . . . . . 8 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) → ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)))
11 reuccatpfxs1lem 41758 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑤𝑋) ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢) ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (𝑊 = (𝑤 prefix (#‘𝑊)) → 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)))
128, 9, 10, 11syl3anc 1366 . . . . . . 7 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) → (𝑊 = (𝑤 prefix (#‘𝑊)) → 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)))
136anim1i 591 . . . . . . . . . . . . . . . 16 (((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) → (𝑊 ∈ Word 𝑉𝑣𝑉))
1413adantr 480 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) → (𝑊 ∈ Word 𝑉𝑣𝑉))
1514ad2antrr 762 . . . . . . . . . . . . . 14 ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → (𝑊 ∈ Word 𝑉𝑣𝑉))
16 lswccats1 13456 . . . . . . . . . . . . . 14 ((𝑊 ∈ Word 𝑉𝑣𝑉) → ( lastS ‘(𝑊 ++ ⟨“𝑣”⟩)) = 𝑣)
1715, 16syl 17 . . . . . . . . . . . . 13 ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → ( lastS ‘(𝑊 ++ ⟨“𝑣”⟩)) = 𝑣)
1817eqcomd 2657 . . . . . . . . . . . 12 ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → 𝑣 = ( lastS ‘(𝑊 ++ ⟨“𝑣”⟩)))
1918s1eqd 13417 . . . . . . . . . . 11 ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → ⟨“𝑣”⟩ = ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩)
2019oveq2d 6706 . . . . . . . . . 10 ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩))
21 id 22 . . . . . . . . . . . 12 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → 𝑤 = (𝑊 ++ ⟨“𝑣”⟩))
22 fveq2 6229 . . . . . . . . . . . . . 14 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ( lastS ‘𝑤) = ( lastS ‘(𝑊 ++ ⟨“𝑣”⟩)))
2322s1eqd 13417 . . . . . . . . . . . . 13 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ⟨“( lastS ‘𝑤)”⟩ = ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩)
2423oveq2d 6706 . . . . . . . . . . . 12 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = (𝑊 ++ ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩))
2521, 24eqeq12d 2666 . . . . . . . . . . 11 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) ↔ (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩)))
2625adantl 481 . . . . . . . . . 10 ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → (𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) ↔ (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩)))
2720, 26mpbird 247 . . . . . . . . 9 ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → 𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩))
28 eleq1 2718 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑤 → (𝑥 ∈ Word 𝑉𝑤 ∈ Word 𝑉))
29 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑤 → (#‘𝑥) = (#‘𝑤))
3029eqeq1d 2653 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑤 → ((#‘𝑥) = ((#‘𝑊) + 1) ↔ (#‘𝑤) = ((#‘𝑊) + 1)))
3128, 30anbi12d 747 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑤 → ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)) ↔ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1))))
3231rspcva 3338 . . . . . . . . . . . . . . . . 17 ((𝑤𝑋 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)))
33 3anass 1059 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word 𝑉𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)) ↔ (𝑊 ∈ Word 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1))))
3433simplbi2com 656 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)) → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1))))
3532, 34syl 17 . . . . . . . . . . . . . . . 16 ((𝑤𝑋 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1))))
3635ex 449 . . . . . . . . . . . . . . 15 (𝑤𝑋 → (∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)) → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)))))
3736com13 88 . . . . . . . . . . . . . 14 (𝑊 ∈ Word 𝑉 → (∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)) → (𝑤𝑋 → (𝑊 ∈ Word 𝑉𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)))))
3837imp 444 . . . . . . . . . . . . 13 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (𝑤𝑋 → (𝑊 ∈ Word 𝑉𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1))))
3938ad2antrr 762 . . . . . . . . . . . 12 ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) → (𝑤𝑋 → (𝑊 ∈ Word 𝑉𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1))))
4039imp 444 . . . . . . . . . . 11 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) → (𝑊 ∈ Word 𝑉𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)))
4140adantr 480 . . . . . . . . . 10 ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → (𝑊 ∈ Word 𝑉𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)))
42 ccats1pfxeqbi 41756 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)) → (𝑊 = (𝑤 prefix (#‘𝑊)) ↔ 𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩)))
4341, 42syl 17 . . . . . . . . 9 ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → (𝑊 = (𝑤 prefix (#‘𝑊)) ↔ 𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩)))
4427, 43mpbird 247 . . . . . . . 8 ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → 𝑊 = (𝑤 prefix (#‘𝑊)))
4544ex 449 . . . . . . 7 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) → (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → 𝑊 = (𝑤 prefix (#‘𝑊))))
4612, 45impbid 202 . . . . . 6 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑤𝑋) → (𝑊 = (𝑤 prefix (#‘𝑊)) ↔ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)))
4746ralrimiva 2995 . . . . 5 ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) → ∀𝑤𝑋 (𝑊 = (𝑤 prefix (#‘𝑊)) ↔ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)))
48 reu6i 3430 . . . . 5 (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑤𝑋 (𝑊 = (𝑤 prefix (#‘𝑊)) ↔ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩))) → ∃!𝑤𝑋 𝑊 = (𝑤 prefix (#‘𝑊)))
495, 47, 48syl2anc 694 . . . 4 ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) → ∃!𝑤𝑋 𝑊 = (𝑤 prefix (#‘𝑊)))
5049ex 449 . . 3 (((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) → (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢)) → ∃!𝑤𝑋 𝑊 = (𝑤 prefix (#‘𝑊))))
5150rexlimdva 3060 . 2 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (∃𝑣𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢)) → ∃!𝑤𝑋 𝑊 = (𝑤 prefix (#‘𝑊))))
524, 51syl5bi 232 1 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑤𝑋 𝑊 = (𝑤 prefix (#‘𝑊))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  ∃!wreu 2943  cfv 5926  (class class class)co 6690  1c1 9975   + caddc 9977  #chash 13157  Word cword 13323   lastS clsw 13324   ++ cconcat 13325  ⟨“cs1 13326   prefix cpfx 41706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-lsw 13332  df-concat 13333  df-s1 13334  df-substr 13335  df-pfx 41707
This theorem is referenced by: (None)
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