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Theorem rusgranumwlkl1 26474
 Description: In a k-regular graph, there are k walks (as word) of length 1 starting at each vertex. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
Assertion
Ref Expression
rusgranumwlkl1 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ∣ (𝑤‘0) = 𝑃}) = 𝐾)
Distinct variable groups:   𝑤,𝐸   𝑤,𝐾   𝑤,𝑃   𝑤,𝑉

Proof of Theorem rusgranumwlkl1
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 rusisusgra 26458 . . . . . . . . . 10 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 USGrph 𝐸)
2 usgrav 25867 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
31, 2syl 17 . . . . . . . . 9 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
4 1nn0 11185 . . . . . . . . . 10 1 ∈ ℕ0
54a1i 11 . . . . . . . . 9 (𝑃𝑉 → 1 ∈ ℕ0)
63, 5anim12i 588 . . . . . . . 8 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 1 ∈ ℕ0))
7 df-3an 1033 . . . . . . . 8 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 1 ∈ ℕ0) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 1 ∈ ℕ0))
86, 7sylibr 223 . . . . . . 7 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 1 ∈ ℕ0))
9 iswwlkn 26212 . . . . . . . 8 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 1 ∈ ℕ0) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ↔ (𝑤 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑤) = (1 + 1))))
10 iswwlk 26211 . . . . . . . . . 10 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑤 ∈ (𝑉 WWalks 𝐸) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)))
11103adant3 1074 . . . . . . . . 9 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 1 ∈ ℕ0) → (𝑤 ∈ (𝑉 WWalks 𝐸) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)))
1211anbi1d 737 . . . . . . . 8 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 1 ∈ ℕ0) → ((𝑤 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑤) = (1 + 1)) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1))))
139, 12bitrd 267 . . . . . . 7 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 1 ∈ ℕ0) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1))))
148, 13syl 17 . . . . . 6 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1))))
1514anbi1d 737 . . . . 5 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ∧ (𝑤‘0) = 𝑃) ↔ (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1)) ∧ (𝑤‘0) = 𝑃)))
16 1p1e2 11011 . . . . . . . . . . 11 (1 + 1) = 2
1716eqeq2i 2622 . . . . . . . . . 10 ((#‘𝑤) = (1 + 1) ↔ (#‘𝑤) = 2)
1817a1i 11 . . . . . . . . 9 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → ((#‘𝑤) = (1 + 1) ↔ (#‘𝑤) = 2))
1918anbi2d 736 . . . . . . . 8 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1)) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = 2)))
20 3anass 1035 . . . . . . . . . . . 12 ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)))
2120a1i 11 . . . . . . . . . . 11 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸))))
22 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑤 = ∅ → (#‘𝑤) = (#‘∅))
23 hash0 13019 . . . . . . . . . . . . . . . 16 (#‘∅) = 0
2422, 23syl6eq 2660 . . . . . . . . . . . . . . 15 (𝑤 = ∅ → (#‘𝑤) = 0)
25 2ne0 10990 . . . . . . . . . . . . . . . . 17 2 ≠ 0
2625nesymi 2839 . . . . . . . . . . . . . . . 16 ¬ 0 = 2
27 eqeq1 2614 . . . . . . . . . . . . . . . 16 ((#‘𝑤) = 0 → ((#‘𝑤) = 2 ↔ 0 = 2))
2826, 27mtbiri 316 . . . . . . . . . . . . . . 15 ((#‘𝑤) = 0 → ¬ (#‘𝑤) = 2)
2924, 28syl 17 . . . . . . . . . . . . . 14 (𝑤 = ∅ → ¬ (#‘𝑤) = 2)
3029necon2ai 2811 . . . . . . . . . . . . 13 ((#‘𝑤) = 2 → 𝑤 ≠ ∅)
3130adantl 481 . . . . . . . . . . . 12 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → 𝑤 ≠ ∅)
3231biantrurd 528 . . . . . . . . . . 11 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → ((𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸))))
33 oveq1 6556 . . . . . . . . . . . . . . . . 17 ((#‘𝑤) = 2 → ((#‘𝑤) − 1) = (2 − 1))
34 2m1e1 11012 . . . . . . . . . . . . . . . . 17 (2 − 1) = 1
3533, 34syl6eq 2660 . . . . . . . . . . . . . . . 16 ((#‘𝑤) = 2 → ((#‘𝑤) − 1) = 1)
3635oveq2d 6565 . . . . . . . . . . . . . . 15 ((#‘𝑤) = 2 → (0..^((#‘𝑤) − 1)) = (0..^1))
3736adantl 481 . . . . . . . . . . . . . 14 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → (0..^((#‘𝑤) − 1)) = (0..^1))
3837raleqdv 3121 . . . . . . . . . . . . 13 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^1){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸))
39 fzo01 12417 . . . . . . . . . . . . . . 15 (0..^1) = {0}
4039raleqi 3119 . . . . . . . . . . . . . 14 (∀𝑖 ∈ (0..^1){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ {0} {(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)
41 c0ex 9913 . . . . . . . . . . . . . . 15 0 ∈ V
42 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑤𝑖) = (𝑤‘0))
43 oveq1 6556 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 0 → (𝑖 + 1) = (0 + 1))
44 0p1e1 11009 . . . . . . . . . . . . . . . . . . 19 (0 + 1) = 1
4543, 44syl6eq 2660 . . . . . . . . . . . . . . . . . 18 (𝑖 = 0 → (𝑖 + 1) = 1)
4645fveq2d 6107 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑤‘(𝑖 + 1)) = (𝑤‘1))
4742, 46preq12d 4220 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → {(𝑤𝑖), (𝑤‘(𝑖 + 1))} = {(𝑤‘0), (𝑤‘1)})
4847eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑖 = 0 → ({(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))
4941, 48ralsn 4169 . . . . . . . . . . . . . 14 (∀𝑖 ∈ {0} {(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)
5040, 49bitri 263 . . . . . . . . . . . . 13 (∀𝑖 ∈ (0..^1){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)
5138, 50syl6bb 275 . . . . . . . . . . . 12 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))
5251anbi2d 736 . . . . . . . . . . 11 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → ((𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)))
5321, 32, 523bitr2d 295 . . . . . . . . . 10 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)))
5453ex 449 . . . . . . . . 9 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → ((#‘𝑤) = 2 → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))))
5554pm5.32rd 670 . . . . . . . 8 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = 2) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ (#‘𝑤) = 2)))
5619, 55bitrd 267 . . . . . . 7 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1)) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ (#‘𝑤) = 2)))
5756anbi1d 737 . . . . . 6 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → ((((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ (#‘𝑤) = 2) ∧ (𝑤‘0) = 𝑃)))
58 anass 679 . . . . . 6 ((((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ (#‘𝑤) = 2) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)))
5957, 58syl6bb 275 . . . . 5 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → ((((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃))))
60 anass 679 . . . . . . 7 (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (𝑤 ∈ Word 𝑉 ∧ ({(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃))))
61 ancom 465 . . . . . . . . 9 (({(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃) ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))
62 df-3an 1033 . . . . . . . . 9 (((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ↔ (((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃) ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))
6361, 62bitr4i 266 . . . . . . . 8 (({(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))
6463anbi2i 726 . . . . . . 7 ((𝑤 ∈ Word 𝑉 ∧ ({(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃))) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)))
6560, 64bitri 263 . . . . . 6 (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)))
6665a1i 11 . . . . 5 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))))
6715, 59, 663bitrd 293 . . . 4 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ∧ (𝑤‘0) = 𝑃) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))))
6867rabbidva2 3162 . . 3 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)})
6968fveq2d 6107 . 2 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ∣ (𝑤‘0) = 𝑃}) = (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}))
70 rusgranumwwlkl1 26473 . 2 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}) = 𝐾)
7169, 70eqtrd 2644 1 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ∣ (𝑤‘0) = 𝑃}) = 𝐾)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173  ∅c0 3874  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  2c2 10947  ℕ0cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   USGrph cusg 25859   WWalks cwwlk 26205   WWalksN cwwlkn 26206   RegUSGrph crusgra 26450 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-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-rmo 2904  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-2o 7448  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-cda 8873  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-xadd 11823  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-nbgra 25949  df-wwlk 26207  df-wwlkn 26208  df-vdgr 26421  df-rgra 26451  df-rusgra 26452 This theorem is referenced by:  rusgranumwlkb1  26481
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