Theorem wwlkextbij 26261

Description: There is a bijection between the extensions of a walk (as word) by an edge and the set of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 21-Aug-2018.)

Assertion

Ref	Expression
wwlkextbij	⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸})

Distinct variable groups:   𝑓,𝐸,𝑤   𝑓,𝑁,𝑤   𝑓,𝑉,𝑤   𝑓,𝑊,𝑤   𝑛,𝐸   𝑛,𝑉   𝑛,𝑊,𝑓

Allowed substitution hint:   𝑁(𝑛)

Proof of Theorem wwlkextbij

Dummy variables 𝑡 𝑥 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6113	. . . 4 ⊢ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∈ V
2	1	a1i 11	. . 3 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∈ V)
3		rabexg 4739	. . 3 ⊢ (((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∈ V → {𝑡 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ∈ V)
4		mptexg 6389	. . 3 ⊢ ({𝑡 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ∈ V → (𝑥 ∈ {𝑡 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)) ∈ V)
5	2, 3, 4	3syl 18	. 2 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑥 ∈ {𝑡 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)) ∈ V)
6		eqid 2610	. . . 4 ⊢ {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)} = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}
7		preq2 4213	. . . . . 6 ⊢ (𝑛 = 𝑝 → {( lastS ‘𝑊), 𝑛} = {( lastS ‘𝑊), 𝑝})
8	7	eleq1d 2672	. . . . 5 ⊢ (𝑛 = 𝑝 → ({( lastS ‘𝑊), 𝑛} ∈ ran 𝐸 ↔ {( lastS ‘𝑊), 𝑝} ∈ ran 𝐸))
9	8	cbvrabv 3172	. . . 4 ⊢ {𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸} = {𝑝 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑝} ∈ ran 𝐸}
10		fveq2 6103	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (#‘𝑡) = (#‘𝑤))
11	10	eqeq1d 2612	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((#‘𝑡) = (𝑁 + 2) ↔ (#‘𝑤) = (𝑁 + 2)))
12		oveq1 6556	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (𝑡 substr ⟨0, (𝑁 + 1)⟩) = (𝑤 substr ⟨0, (𝑁 + 1)⟩))
13	12	eqeq1d 2612	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
14		fveq2 6103	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → ( lastS ‘𝑡) = ( lastS ‘𝑤))
15	14	preq2d 4219	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → {( lastS ‘𝑊), ( lastS ‘𝑡)} = {( lastS ‘𝑊), ( lastS ‘𝑤)})
16	15	eleq1d 2672	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ({( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))
17	11, 13, 16	3anbi123d 1391	. . . . . 6 ⊢ (𝑡 = 𝑤 → (((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸) ↔ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)))
18	17	cbvrabv 3172	. . . . 5 ⊢ {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}
19		mpteq1 4665	. . . . 5 ⊢ ({𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)} → (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)) = (𝑥 ∈ {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)))
20	18, 19	ax-mp 5	. . . 4 ⊢ (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)) = (𝑥 ∈ {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥))
21	6, 9, 20	wwlkextbij0 26260	. . 3 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)):{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸})
22		eqid 2610	. . . . . . 7 ⊢ {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} = {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)}
23	22	wwlkextwrd 26256	. . . . . 6 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} = {𝑡 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)})
24	23	eqcomd 2616	. . . . 5 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → {𝑡 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} = {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)})
25	24	mpteq1d 4666	. . . 4 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑥 ∈ {𝑡 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)) = (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)))
26	6	wwlkextwrd 26256	. . . . 5 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)})
27	26	eqcomd 2616	. . . 4 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)} = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)})
28		eqidd 2611	. . . 4 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → {𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸} = {𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸})
29	25, 27, 28	f1oeq123d 6046	. . 3 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑥 ∈ {𝑡 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)):{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸} ↔ (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)):{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}))
30	21, 29	mpbird 246	. 2 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑥 ∈ {𝑡 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)):{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸})
31		f1oeq1 6040	. . 3 ⊢ (𝑓 = (𝑥 ∈ {𝑡 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)) → (𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸} ↔ (𝑥 ∈ {𝑡 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)):{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}))
32	31	spcegv 3267	. 2 ⊢ ((𝑥 ∈ {𝑡 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)) ∈ V → ((𝑥 ∈ {𝑡 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)} ↦ ( lastS ‘𝑥)):{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸} → ∃𝑓 𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}))
33	5, 30, 32	sylc 63	1 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸})

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {crab 2900  Vcvv 3173  {cpr 4127  ⟨cop 4131   ↦ cmpt 4643  ran crn 5039  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  #chash 12979  Word cword 13146   lastS clsw 13147   substr csubstr 13150   WWalksN cwwlkn 26206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-wwlk 26207  df-wwlkn 26208

This theorem is referenced by:  wwlkexthasheq  26262