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Theorem clwwlknprop 26300
 Description: Properties of a closed walk of a fixed length as word. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
Assertion
Ref Expression
clwwlknprop (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))

Proof of Theorem clwwlknprop
Dummy variables 𝑒 𝑛 𝑝 𝑡 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ne0i 3880 . . 3 (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ClWWalksN 𝐸)‘𝑁) ≠ ∅)
2 ndmfv 6128 . . . 4 𝑁 ∈ dom (𝑉 ClWWalksN 𝐸) → ((𝑉 ClWWalksN 𝐸)‘𝑁) = ∅)
32necon1ai 2809 . . 3 (((𝑉 ClWWalksN 𝐸)‘𝑁) ≠ ∅ → 𝑁 ∈ dom (𝑉 ClWWalksN 𝐸))
4 oveq12 6558 . . . . . . . . . . 11 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 ClWWalks 𝑒) = (𝑉 ClWWalks 𝐸))
54rabeqdv 3167 . . . . . . . . . 10 ((𝑣 = 𝑉𝑒 = 𝐸) → {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛} = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})
65mpteq2dv 4673 . . . . . . . . 9 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}))
7 df-clwwlkn 26280 . . . . . . . . 9 ClWWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}))
8 nn0ex 11175 . . . . . . . . . 10 0 ∈ V
98mptex 6390 . . . . . . . . 9 (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}) ∈ V
106, 7, 9ovmpt2a 6689 . . . . . . . 8 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ClWWalksN 𝐸) = (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}))
11103adant3 1074 . . . . . . 7 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → (𝑉 ClWWalksN 𝐸) = (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}))
1211dmeqd 5248 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → dom (𝑉 ClWWalksN 𝐸) = dom (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}))
1312eleq2d 2673 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → (𝑁 ∈ dom (𝑉 ClWWalksN 𝐸) ↔ 𝑁 ∈ dom (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})))
14 dmopabss 5258 . . . . . . . 8 dom {⟨𝑛, 𝑡⟩ ∣ (𝑛 ∈ ℕ0𝑡 = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})} ⊆ ℕ0
1514sseli 3564 . . . . . . 7 (𝑁 ∈ dom {⟨𝑛, 𝑡⟩ ∣ (𝑛 ∈ ℕ0𝑡 = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})} → 𝑁 ∈ ℕ0)
16 clwwlkn 26295 . . . . . . . . . . . . . . 15 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑉 ClWWalksN 𝐸)‘𝑁) = {𝑝 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑝) = 𝑁})
17163expa 1257 . . . . . . . . . . . . . 14 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0) → ((𝑉 ClWWalksN 𝐸)‘𝑁) = {𝑝 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑝) = 𝑁})
1817eleq2d 2673 . . . . . . . . . . . . 13 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ 𝑃 ∈ {𝑝 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑝) = 𝑁}))
19 fveq2 6103 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = 𝑃 → (#‘𝑝) = (#‘𝑃))
2019eqeq1d 2612 . . . . . . . . . . . . . . . . . . 19 (𝑝 = 𝑃 → ((#‘𝑝) = 𝑁 ↔ (#‘𝑃) = 𝑁))
2120elrab 3331 . . . . . . . . . . . . . . . . . 18 (𝑃 ∈ {𝑝 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑝) = 𝑁} ↔ (𝑃 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑃) = 𝑁))
22 clwwlkprop 26298 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉))
23 simpl3 1059 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ (#‘𝑃) = 𝑁) → 𝑃 ∈ Word 𝑉)
24 lencl 13179 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑃 ∈ Word 𝑉 → (#‘𝑃) ∈ ℕ0)
25 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑃) = 𝑁 → ((#‘𝑃) ∈ ℕ0𝑁 ∈ ℕ0))
2624, 25syl5ibcom 234 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃 ∈ Word 𝑉 → ((#‘𝑃) = 𝑁𝑁 ∈ ℕ0))
27263ad2ant3 1077 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) → ((#‘𝑃) = 𝑁𝑁 ∈ ℕ0))
2827impac 649 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ (#‘𝑃) = 𝑁) → (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))
2923, 28jca 553 . . . . . . . . . . . . . . . . . . 19 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ (#‘𝑃) = 𝑁) → (𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))
3022, 29sylan 487 . . . . . . . . . . . . . . . . . 18 ((𝑃 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑃) = 𝑁) → (𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))
3121, 30sylbi 206 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ {𝑝 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑝) = 𝑁} → (𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))
3231anim2i 591 . . . . . . . . . . . . . . . 16 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ {𝑝 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑝) = 𝑁}) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))))
33 3anass 1035 . . . . . . . . . . . . . . . 16 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))))
3432, 33sylibr 223 . . . . . . . . . . . . . . 15 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ {𝑝 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑝) = 𝑁}) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))
3534ex 449 . . . . . . . . . . . . . 14 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ {𝑝 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑝) = 𝑁} → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))))
3635adantr 480 . . . . . . . . . . . . 13 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0) → (𝑃 ∈ {𝑝 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑝) = 𝑁} → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))))
3718, 36sylbid 229 . . . . . . . . . . . 12 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))))
3837ex 449 . . . . . . . . . . 11 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ ℕ0 → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))))
3938com23 84 . . . . . . . . . 10 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℕ0 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))))
4039a1d 25 . . . . . . . . 9 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (((𝑉 ClWWalksN 𝐸)‘𝑁) ≠ ∅ → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℕ0 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))))))
417mpt2ndm0 6773 . . . . . . . . . 10 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ClWWalksN 𝐸) = ∅)
42 fveq1 6102 . . . . . . . . . . 11 ((𝑉 ClWWalksN 𝐸) = ∅ → ((𝑉 ClWWalksN 𝐸)‘𝑁) = (∅‘𝑁))
43 0fv 6137 . . . . . . . . . . 11 (∅‘𝑁) = ∅
4442, 43syl6eq 2660 . . . . . . . . . 10 ((𝑉 ClWWalksN 𝐸) = ∅ → ((𝑉 ClWWalksN 𝐸)‘𝑁) = ∅)
45 eqneqall 2793 . . . . . . . . . 10 (((𝑉 ClWWalksN 𝐸)‘𝑁) = ∅ → (((𝑉 ClWWalksN 𝐸)‘𝑁) ≠ ∅ → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℕ0 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))))))
4641, 44, 453syl 18 . . . . . . . . 9 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (((𝑉 ClWWalksN 𝐸)‘𝑁) ≠ ∅ → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℕ0 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))))))
4740, 46pm2.61i 175 . . . . . . . 8 (((𝑉 ClWWalksN 𝐸)‘𝑁) ≠ ∅ → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℕ0 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))))
481, 47mpcom 37 . . . . . . 7 (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℕ0 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))))
4915, 48syl5com 31 . . . . . 6 (𝑁 ∈ dom {⟨𝑛, 𝑡⟩ ∣ (𝑛 ∈ ℕ0𝑡 = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})} → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))))
50 df-mpt 4645 . . . . . . 7 (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}) = {⟨𝑛, 𝑡⟩ ∣ (𝑛 ∈ ℕ0𝑡 = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})}
5150dmeqi 5247 . . . . . 6 dom (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}) = dom {⟨𝑛, 𝑡⟩ ∣ (𝑛 ∈ ℕ0𝑡 = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})}
5249, 51eleq2s 2706 . . . . 5 (𝑁 ∈ dom (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))))
5313, 52syl6bi 242 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → (𝑁 ∈ dom (𝑉 ClWWalksN 𝐸) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))))
54 3ianor 1048 . . . . 5 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) ↔ (¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V ∨ ¬ (𝑉 ∈ V ∧ 𝐸 ∈ V)))
55 df-3or 1032 . . . . . 6 ((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V ∨ ¬ (𝑉 ∈ V ∧ 𝐸 ∈ V)) ↔ ((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V) ∨ ¬ (𝑉 ∈ V ∧ 𝐸 ∈ V)))
56 ianor 508 . . . . . . . 8 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) ↔ (¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V))
5741dmeqd 5248 . . . . . . . . . . 11 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → dom (𝑉 ClWWalksN 𝐸) = dom ∅)
5857eleq2d 2673 . . . . . . . . . 10 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ dom (𝑉 ClWWalksN 𝐸) ↔ 𝑁 ∈ dom ∅))
59 dm0 5260 . . . . . . . . . . 11 dom ∅ = ∅
6059eleq2i 2680 . . . . . . . . . 10 (𝑁 ∈ dom ∅ ↔ 𝑁 ∈ ∅)
6158, 60syl6bb 275 . . . . . . . . 9 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ dom (𝑉 ClWWalksN 𝐸) ↔ 𝑁 ∈ ∅))
62 noel 3878 . . . . . . . . . 10 ¬ 𝑁 ∈ ∅
6362pm2.21i 115 . . . . . . . . 9 (𝑁 ∈ ∅ → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))))
6461, 63syl6bi 242 . . . . . . . 8 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ dom (𝑉 ClWWalksN 𝐸) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))))
6556, 64sylbir 224 . . . . . . 7 ((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V) → (𝑁 ∈ dom (𝑉 ClWWalksN 𝐸) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))))
6665, 64jaoi 393 . . . . . 6 (((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V) ∨ ¬ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → (𝑁 ∈ dom (𝑉 ClWWalksN 𝐸) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))))
6755, 66sylbi 206 . . . . 5 ((¬ 𝑉 ∈ V ∨ ¬ 𝐸 ∈ V ∨ ¬ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → (𝑁 ∈ dom (𝑉 ClWWalksN 𝐸) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))))
6854, 67sylbi 206 . . . 4 (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → (𝑁 ∈ dom (𝑉 ClWWalksN 𝐸) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))))
6953, 68pm2.61i 175 . . 3 (𝑁 ∈ dom (𝑉 ClWWalksN 𝐸) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))))
701, 3, 693syl 18 . 2 (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))))
7170pm2.43i 50 1 (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173  ∅c0 3874  {copab 4642   ↦ cmpt 4643  dom cdm 5038  ‘cfv 5804  (class class class)co 6549  ℕ0cn0 11169  #chash 12979  Word cword 13146   ClWWalks cclwwlk 26276   ClWWalksN cclwwlkn 26277 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-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-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-clwwlk 26279  df-clwwlkn 26280 This theorem is referenced by:  clwwlknndef  26301  clwwlknimp  26304  clwwlknwwlkncl  26328  clwwlkext2edg  26330  wwlksubclwwlk  26332  clwwnisshclwwn  26337  eleclclwwlknlem1  26345  eleclclwwlknlem2  26346  clwwlknscsh  26347  erclwwlkneqlen  26352  erclwwlknref  26353  erclwwlknsym  26354  erclwwlkntr  26355  hashecclwwlkn1  26361  usghashecclwwlk  26362  clwlkfoclwwlk  26372  extwwlkfablem2  26605  numclwlk2lem2f  26630  numclwlk2lem2f1o  26632
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