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Theorem clwlkclwwlkfo 27159
 Description: 𝐹 is a function from the nonempty closed walks onto the closed walks as word in a simple pseudograph. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 2-May-2021.) (Revised by AV, 25-May-2022.)
Hypotheses
Ref Expression
clwlkclwwlkf.c 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
clwlkclwwlkf.f 𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩))
Assertion
Ref Expression
clwlkclwwlkfo (𝐺 ∈ USPGraph → 𝐹:𝐶onto→(ClWWalks‘𝐺))
Distinct variable groups:   𝑤,𝐺,𝑐   𝐶,𝑐,𝑤   𝐹,𝑐,𝑤

Proof of Theorem clwlkclwwlkfo
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 clwlkclwwlkf.c . . 3 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
2 clwlkclwwlkf.f . . 3 𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩))
31, 2clwlkclwwlkf 27158 . 2 (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4 clwwlkgt0 27136 . . . . . 6 (𝑤 ∈ (ClWWalks‘𝐺) → 0 < (♯‘𝑤))
5 eqid 2771 . . . . . . . 8 (Vtx‘𝐺) = (Vtx‘𝐺)
65clwwlkbp 27135 . . . . . . 7 (𝑤 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑤 ≠ ∅))
7 lencl 13520 . . . . . . . . . . . 12 (𝑤 ∈ Word (Vtx‘𝐺) → (♯‘𝑤) ∈ ℕ0)
87nn0zd 11682 . . . . . . . . . . 11 (𝑤 ∈ Word (Vtx‘𝐺) → (♯‘𝑤) ∈ ℤ)
9 zgt0ge1 11633 . . . . . . . . . . 11 ((♯‘𝑤) ∈ ℤ → (0 < (♯‘𝑤) ↔ 1 ≤ (♯‘𝑤)))
108, 9syl 17 . . . . . . . . . 10 (𝑤 ∈ Word (Vtx‘𝐺) → (0 < (♯‘𝑤) ↔ 1 ≤ (♯‘𝑤)))
1110biimpd 219 . . . . . . . . 9 (𝑤 ∈ Word (Vtx‘𝐺) → (0 < (♯‘𝑤) → 1 ≤ (♯‘𝑤)))
1211anc2li 545 . . . . . . . 8 (𝑤 ∈ Word (Vtx‘𝐺) → (0 < (♯‘𝑤) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))))
13123ad2ant2 1128 . . . . . . 7 ((𝐺 ∈ V ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑤 ≠ ∅) → (0 < (♯‘𝑤) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))))
146, 13syl 17 . . . . . 6 (𝑤 ∈ (ClWWalks‘𝐺) → (0 < (♯‘𝑤) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))))
154, 14mpd 15 . . . . 5 (𝑤 ∈ (ClWWalks‘𝐺) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
1615adantl 467 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (ClWWalks‘𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
17 eqid 2771 . . . . . . . . 9 (iEdg‘𝐺) = (iEdg‘𝐺)
185, 17clwlkclwwlk2 27153 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalks‘𝐺)))
19 df-br 4787 . . . . . . . . . 10 (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
20 simpr2 1235 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → 𝑤 ∈ Word (Vtx‘𝐺))
21 simpr3 1237 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → 1 ≤ (♯‘𝑤))
22 simpl 468 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
231clwlkclwwlkfolem 27157 . . . . . . . . . . . . . 14 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
2420, 21, 22, 23syl3anc 1476 . . . . . . . . . . . . 13 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
25233expa 1111 . . . . . . . . . . . . . . . . . . 19 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
26 ovex 6823 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)⟩) ∈ V
27 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (2nd𝑐) = (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
2827fveq2d 6336 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (♯‘(2nd𝑐)) = (♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
2928oveq1d 6808 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((♯‘(2nd𝑐)) − 1) = ((♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1))
3029opeq2d 4546 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ⟨0, ((♯‘(2nd𝑐)) − 1)⟩ = ⟨0, ((♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1)⟩)
3127, 30oveq12d 6811 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩) = ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, ((♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1)⟩))
32 vex 3354 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓 ∈ V
33 ovex 6823 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ++ ⟨“(𝑤‘0)”⟩) ∈ V
3432, 33op2nd 7324 . . . . . . . . . . . . . . . . . . . . . 22 (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = (𝑤 ++ ⟨“(𝑤‘0)”⟩)
3534fveq2i 6335 . . . . . . . . . . . . . . . . . . . . . . . 24 (♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩))
3635oveq1i 6803 . . . . . . . . . . . . . . . . . . . . . . 23 ((♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1) = ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)
3736opeq2i 4543 . . . . . . . . . . . . . . . . . . . . . 22 ⟨0, ((♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1)⟩ = ⟨0, ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)⟩
3834, 37oveq12i 6805 . . . . . . . . . . . . . . . . . . . . 21 ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, ((♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)⟩)
3931, 38syl6eq 2821 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)⟩))
4039, 2fvmptg 6422 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ∧ ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)⟩) ∈ V) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)⟩))
4125, 26, 40sylancl 574 . . . . . . . . . . . . . . . . . 18 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)⟩))
42 wrdlenccats1lenm1 13603 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ Word (Vtx‘𝐺) → ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1) = (♯‘𝑤))
4342ad2antrr 705 . . . . . . . . . . . . . . . . . . . 20 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1) = (♯‘𝑤))
4443opeq2d 4546 . . . . . . . . . . . . . . . . . . 19 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨0, ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)⟩ = ⟨0, (♯‘𝑤)⟩)
4544oveq2d 6809 . . . . . . . . . . . . . . . . . 18 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑤)⟩))
46 simpll 750 . . . . . . . . . . . . . . . . . . 19 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 𝑤 ∈ Word (Vtx‘𝐺))
47 simpl 468 . . . . . . . . . . . . . . . . . . . . 21 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
48 wrdsymb1 13539 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑤‘0) ∈ (Vtx‘𝐺))
4947, 48syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝑤‘0) ∈ (Vtx‘𝐺))
5049s1cld 13583 . . . . . . . . . . . . . . . . . . 19 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺))
51 eqidd 2772 . . . . . . . . . . . . . . . . . . 19 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (♯‘𝑤) = (♯‘𝑤))
52 swrdccatid 13706 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (♯‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑤)⟩) = 𝑤)
5346, 50, 51, 52syl3anc 1476 . . . . . . . . . . . . . . . . . 18 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑤)⟩) = 𝑤)
5441, 45, 533eqtrrd 2810 . . . . . . . . . . . . . . . . 17 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
5554ex 397 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
56553adant1 1124 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
5756ad2antlr 706 . . . . . . . . . . . . . 14 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
58 fveq2 6332 . . . . . . . . . . . . . . . . 17 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐹𝑐) = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
5958eqeq2d 2781 . . . . . . . . . . . . . . . 16 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝑤 = (𝐹𝑐) ↔ 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
6059imbi2d 329 . . . . . . . . . . . . . . 15 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
6160adantl 467 . . . . . . . . . . . . . 14 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
6257, 61mpbird 247 . . . . . . . . . . . . 13 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹𝑐)))
6324, 62rspcimedv 3462 . . . . . . . . . . . 12 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
6463ex 397 . . . . . . . . . . 11 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐))))
6564pm2.43b 55 . . . . . . . . . 10 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
6619, 65syl5bi 232 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
6766exlimdv 2013 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
6818, 67sylbird 250 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑤 ∈ (ClWWalks‘𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
69683expib 1116 . . . . . 6 (𝐺 ∈ USPGraph → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑤 ∈ (ClWWalks‘𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐))))
7069com23 86 . . . . 5 (𝐺 ∈ USPGraph → (𝑤 ∈ (ClWWalks‘𝐺) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → ∃𝑐𝐶 𝑤 = (𝐹𝑐))))
7170imp 393 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (ClWWalks‘𝐺)) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
7216, 71mpd 15 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (ClWWalks‘𝐺)) → ∃𝑐𝐶 𝑤 = (𝐹𝑐))
7372ralrimiva 3115 . 2 (𝐺 ∈ USPGraph → ∀𝑤 ∈ (ClWWalks‘𝐺)∃𝑐𝐶 𝑤 = (𝐹𝑐))
74 dffo3 6517 . 2 (𝐹:𝐶onto→(ClWWalks‘𝐺) ↔ (𝐹:𝐶⟶(ClWWalks‘𝐺) ∧ ∀𝑤 ∈ (ClWWalks‘𝐺)∃𝑐𝐶 𝑤 = (𝐹𝑐)))
753, 73, 74sylanbrc 572 1 (𝐺 ∈ USPGraph → 𝐹:𝐶onto→(ClWWalks‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631  ∃wex 1852   ∈ wcel 2145   ≠ wne 2943  ∀wral 3061  ∃wrex 3062  {crab 3065  Vcvv 3351  ∅c0 4063  ⟨cop 4322   class class class wbr 4786   ↦ cmpt 4863  ⟶wf 6027  –onto→wfo 6029  ‘cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  0cc0 10138  1c1 10139   < clt 10276   ≤ cle 10277   − cmin 10468  ℤcz 11579  ♯chash 13321  Word cword 13487   ++ cconcat 13489  ⟨“cs1 13490   substr csubstr 13491  Vtxcvtx 26095  iEdgciedg 26096  USPGraphcuspgr 26265  ClWalkscclwlks 26901  ClWWalkscclwwlk 27131 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-rp 12036  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-lsw 13496  df-concat 13497  df-s1 13498  df-substr 13499  df-edg 26161  df-uhgr 26174  df-upgr 26198  df-uspgr 26267  df-wlks 26730  df-clwlks 26902  df-clwwlk 27132 This theorem is referenced by:  clwlkclwwlkf1o  27161
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