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Theorem upgriswlk 29673
Description: Properties of a pair of functions to be a walk in a pseudograph. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 28-Oct-2021.)
Hypotheses
Ref Expression
upgriswlk.v 𝑉 = (Vtx‘𝐺)
upgriswlk.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
upgriswlk (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
Distinct variable groups:   𝑘,𝐺   𝑘,𝐹   𝑘,𝐼   𝑃,𝑘   𝑘,𝑉

Proof of Theorem upgriswlk
StepHypRef Expression
1 upgriswlk.v . . 3 𝑉 = (Vtx‘𝐺)
2 upgriswlk.i . . 3 𝐼 = (iEdg‘𝐺)
31, 2iswlkg 29645 . 2 (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))
4 df-ifp 1063 . . . . . . 7 (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ↔ (((𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}) ∨ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))))
5 dfsn2 4643 . . . . . . . . . . . . 13 {(𝑃𝑘)} = {(𝑃𝑘), (𝑃𝑘)}
6 preq2 4738 . . . . . . . . . . . . 13 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃𝑘), (𝑃𝑘)} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
75, 6eqtrid 2786 . . . . . . . . . . . 12 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃𝑘)} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
87eqeq2d 2745 . . . . . . . . . . 11 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘)} ↔ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
98biimpa 476 . . . . . . . . . 10 (((𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}) → (𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
109a1d 25 . . . . . . . . 9 (((𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}) → (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
11 eqid 2734 . . . . . . . . . . . . . 14 (Edg‘𝐺) = (Edg‘𝐺)
122, 11upgredginwlk 29668 . . . . . . . . . . . . 13 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom 𝐼) → (𝑘 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺)))
1312adantrr 717 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) → (𝑘 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺)))
1413imp 406 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺))
15 simp-4l 783 . . . . . . . . . . . . . 14 (((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))) → 𝐺 ∈ UPGraph)
16 simpr 484 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺)) → (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺))
1716adantr 480 . . . . . . . . . . . . . 14 (((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))) → (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺))
18 simpr 484 . . . . . . . . . . . . . . 15 ((¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))
1918adantl 481 . . . . . . . . . . . . . 14 (((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))) → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))
20 fvexd 6921 . . . . . . . . . . . . . . . . 17 (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) → (𝑃𝑘) ∈ V)
21 fvexd 6921 . . . . . . . . . . . . . . . . 17 (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) → (𝑃‘(𝑘 + 1)) ∈ V)
22 neqne 2945 . . . . . . . . . . . . . . . . 17 (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))
2320, 21, 223jca 1127 . . . . . . . . . . . . . . . 16 (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) → ((𝑃𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
2423adantr 480 . . . . . . . . . . . . . . 15 ((¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → ((𝑃𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
2524adantl 481 . . . . . . . . . . . . . 14 (((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))) → ((𝑃𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
261, 11upgredgpr 29173 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ∧ ((𝑃𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))) → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = (𝐼‘(𝐹𝑘)))
2715, 17, 19, 25, 26syl31anc 1372 . . . . . . . . . . . . 13 (((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))) → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = (𝐼‘(𝐹𝑘)))
2827eqcomd 2740 . . . . . . . . . . . 12 (((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))) → (𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
2928exp31 419 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺) → ((¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → (𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
3014, 29mpd 15 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → (𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
3130com12 32 . . . . . . . . 9 ((¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
3210, 31jaoi 857 . . . . . . . 8 ((((𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}) ∨ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))) → (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
3332com12 32 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((((𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}) ∨ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))) → (𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
344, 33biimtrid 242 . . . . . 6 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → (𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
35 ifpprsnss 4768 . . . . . 6 ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
3634, 35impbid1 225 . . . . 5 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ↔ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
3736ralbidva 3173 . . . 4 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉)) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ↔ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
3837pm5.32da 579 . . 3 (𝐺 ∈ UPGraph → (((𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))) ↔ ((𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
39 df-3an 1088 . . 3 ((𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))) ↔ ((𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))))
40 df-3an 1088 . . 3 ((𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ↔ ((𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
4138, 39, 403bitr4g 314 . 2 (𝐺 ∈ UPGraph → ((𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))) ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
423, 41bitrd 279 1 (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  if-wif 1062  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  Vcvv 3477  wss 3962  {csn 4630  {cpr 4632   class class class wbr 5147  dom cdm 5688  wf 6558  cfv 6562  (class class class)co 7430  0cc0 11152  1c1 11153   + caddc 11155  ...cfz 13543  ..^cfzo 13690  chash 14365  Word cword 14548  Vtxcvtx 29027  iEdgciedg 29028  Edgcedg 29078  UPGraphcupgr 29111  Walkscwlks 29628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-hash 14366  df-word 14549  df-edg 29079  df-uhgr 29089  df-upgr 29113  df-wlks 29631
This theorem is referenced by:  upgrwlkedg  29674  upgrwlkcompim  29675  upgrwlkvtxedg  29677  upgr2wlk  29700  upgrtrls  29733  upgristrl  29734  upgrwlkdvde  29769  usgr2wlkneq  29788  isclwlkupgr  29810  uspgrn2crct  29837  wlkiswwlks1  29896  wlkiswwlks2  29904  wlkiswwlksupgr2  29906  wlk2v2e  30185  upgriseupth  30235
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