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Theorem upgriswlk 29659
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 29631 . 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 1064 . . . . . . 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 4639 . . . . . . . . . . . . 13 {(𝑃𝑘)} = {(𝑃𝑘), (𝑃𝑘)}
6 preq2 4734 . . . . . . . . . . . . 13 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃𝑘), (𝑃𝑘)} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
75, 6eqtrid 2789 . . . . . . . . . . . 12 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃𝑘)} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
87eqeq2d 2748 . . . . . . . . . . 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 2737 . . . . . . . . . . . . . 14 (Edg‘𝐺) = (Edg‘𝐺)
122, 11upgredginwlk 29654 . . . . . . . . . . . . 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 2948 . . . . . . . . . . . . . . . . 17 (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))
2320, 21, 223jca 1129 . . . . . . . . . . . . . . . 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 29159 . . . . . . . . . . . . . 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 1375 . . . . . . . . . . . . 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 2743 . . . . . . . . . . . 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 858 . . . . . . . 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 4764 . . . . . 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 3176 . . . 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 1089 . . 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 1089 . . 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 848  if-wif 1063  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480  wss 3951  {csn 4626  {cpr 4628   class class class wbr 5143  dom cdm 5685  wf 6557  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156   + caddc 11158  ...cfz 13547  ..^cfzo 13694  chash 14369  Word cword 14552  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064  UPGraphcupgr 29097  Walkscwlks 29614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-edg 29065  df-uhgr 29075  df-upgr 29099  df-wlks 29617
This theorem is referenced by:  upgrwlkedg  29660  upgrwlkcompim  29661  upgrwlkvtxedg  29663  upgr2wlk  29686  upgrtrls  29719  upgristrl  29720  upgrwlkdvde  29757  usgr2wlkneq  29776  isclwlkupgr  29798  uspgrn2crct  29828  wlkiswwlks1  29887  wlkiswwlks2  29895  wlkiswwlksupgr2  29897  wlk2v2e  30176  upgriseupth  30226
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