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Theorem upgriswlk 26440
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 26413 . 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 1012 . . . . . . 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 4168 . . . . . . . . . . . . 13 {(𝑃𝑘)} = {(𝑃𝑘), (𝑃𝑘)}
6 preq2 4246 . . . . . . . . . . . . 13 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃𝑘), (𝑃𝑘)} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
75, 6syl5eq 2667 . . . . . . . . . . . 12 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃𝑘)} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
87eqeq2d 2631 . . . . . . . . . . 11 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘)} ↔ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
98biimpa 501 . . . . . . . . . 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 2621 . . . . . . . . . . . . . 14 (Edg‘𝐺) = (Edg‘𝐺)
122, 11upgredginwlk 26435 . . . . . . . . . . . . 13 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom 𝐼) → (𝑘 ∈ (0..^(#‘𝐹)) → (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺)))
1312adantrr 752 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉)) → (𝑘 ∈ (0..^(#‘𝐹)) → (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺)))
1413imp 445 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(#‘𝐹))) → (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺))
15 simp-4l 805 . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(#‘𝐹))) ∧ (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺)) → (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺))
1716adantr 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)))) → (𝐼‘(𝐹𝑘)) ∈ (Edg‘𝐺))
18 simpr 477 . . . . . . . . . . . . . . 15 ((¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))
1918adantl 482 . . . . . . . . . . . . . 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 6170 . . . . . . . . . . . . . . . . 17 (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) → (𝑃𝑘) ∈ V)
21 fvexd 6170 . . . . . . . . . . . . . . . . 17 (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) → (𝑃‘(𝑘 + 1)) ∈ V)
22 neqne 2798 . . . . . . . . . . . . . . . . 17 (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))
2320, 21, 223jca 1240 . . . . . . . . . . . . . . . 16 (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) → ((𝑃𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
2423adantr 481 . . . . . . . . . . . . . . 15 ((¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → ((𝑃𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
2524adantl 482 . . . . . . . . . . . . . 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 25966 . . . . . . . . . . . . . 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 1326 . . . . . . . . . . . . 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 2627 . . . . . . . . . . . 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 629 . . . . . . . . . . 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 394 . . . . . . . 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, 33syl5bi 232 . . . . . 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 4276 . . . . . 6 ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
3634, 35impbid1 215 . . . . 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 2981 . . . 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 672 . . 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 1038 . . 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 1038 . . 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 303 . 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 268 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 196  wo 383  wa 384  if-wif 1011  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2908  Vcvv 3190  wss 3560  {csn 4155  {cpr 4157   class class class wbr 4623  dom cdm 5084  wf 5853  cfv 5857  (class class class)co 6615  0cc0 9896  1c1 9897   + caddc 9899  ...cfz 12284  ..^cfzo 12422  #chash 13073  Word cword 13246  Vtxcvtx 25808  iEdgciedg 25809  Edgcedg 25873   UPGraph cupgr 25905  Walkscwlks 26396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-xnn0 11324  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-hash 13074  df-word 13254  df-edg 25874  df-uhgr 25883  df-upgr 25907  df-wlks 26399
This theorem is referenced by:  upgrwlkedg  26441  upgrwlkcompim  26442  upgrwlkvtxedg  26444  upgr2wlk  26467  upgrtrls  26501  upgristrl  26502  upgrwlkdvde  26536  usgr2wlkneq  26555  isclwlkupgr  26577  uspgrn2crct  26603  wlkiswwlks1  26656  wlkiswwlks2  26664  wlkiswwlksupgr2  26666  wlk2v2e  26917  upgriseupth  26967
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