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Theorem wpthswwlks2on 26729
 Description: For two different vertices, a walk of length 2 between these vertices is a simple path of length 2 between these vertices in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 13-May-2021.)
Hypothesis
Ref Expression
wpthswwlks2on.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
wpthswwlks2on ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))

Proof of Theorem wpthswwlks2on
Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wpthswwlks2on.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
21wwlknon 26618 . . . . . . 7 ((𝐴𝑉𝐵𝑉) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
323ad2ant2 1081 . . . . . 6 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
43anbi1d 740 . . . . 5 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ ((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
5 3anass 1040 . . . . . . 7 ((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
65anbi1i 730 . . . . . 6 (((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ ((𝑤 ∈ (2 WWalksN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
7 anass 680 . . . . . 6 (((𝑤 ∈ (2 WWalksN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
86, 7bitri 264 . . . . 5 (((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
94, 8syl6bb 276 . . . 4 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))))
109rabbidva2 3174 . . 3 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = {𝑤 ∈ (2 WWalksN 𝐺) ∣ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)})
11 usgrupgr 25977 . . . . . . . . . . 11 (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph )
12 wlklnwwlknupgr 26648 . . . . . . . . . . 11 (𝐺 ∈ UPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2) ↔ 𝑤 ∈ (2 WWalksN 𝐺)))
1311, 12syl 17 . . . . . . . . . 10 (𝐺 ∈ USGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2) ↔ 𝑤 ∈ (2 WWalksN 𝐺)))
1413bicomd 213 . . . . . . . . 9 (𝐺 ∈ USGraph → (𝑤 ∈ (2 WWalksN 𝐺) ↔ ∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)))
15143ad2ant1 1080 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑤 ∈ (2 WWalksN 𝐺) ↔ ∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)))
16 simprl 793 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → 𝑓(Walks‘𝐺)𝑤)
17 simprl 793 . . . . . . . . . . . . . . 15 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (𝑤‘0) = 𝐴)
1817adantr 481 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (𝑤‘0) = 𝐴)
19 fveq2 6150 . . . . . . . . . . . . . . . 16 ((#‘𝑓) = 2 → (𝑤‘(#‘𝑓)) = (𝑤‘2))
2019ad2antll 764 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (𝑤‘(#‘𝑓)) = (𝑤‘2))
21 simprr 795 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (𝑤‘2) = 𝐵)
2221adantr 481 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (𝑤‘2) = 𝐵)
2320, 22eqtrd 2655 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (𝑤‘(#‘𝑓)) = 𝐵)
24 simpll2 1099 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (𝐴𝑉𝐵𝑉))
25 vex 3189 . . . . . . . . . . . . . . . 16 𝑓 ∈ V
26 vex 3189 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
2725, 26pm3.2i 471 . . . . . . . . . . . . . . 15 (𝑓 ∈ V ∧ 𝑤 ∈ V)
281iswlkon 26429 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝐵𝑉) ∧ (𝑓 ∈ V ∧ 𝑤 ∈ V)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤 ↔ (𝑓(Walks‘𝐺)𝑤 ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘(#‘𝑓)) = 𝐵)))
2924, 27, 28sylancl 693 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤 ↔ (𝑓(Walks‘𝐺)𝑤 ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘(#‘𝑓)) = 𝐵)))
3016, 18, 23, 29mpbir3and 1243 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → 𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤)
31 simpll1 1098 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → 𝐺 ∈ USGraph )
32 simprr 795 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (#‘𝑓) = 2)
33 simpll3 1100 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → 𝐴𝐵)
34 usgr2wlkspth 26531 . . . . . . . . . . . . . 14 ((𝐺 ∈ USGraph ∧ (#‘𝑓) = 2 ∧ 𝐴𝐵) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
3531, 32, 33, 34syl3anc 1323 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
3630, 35mpbid 222 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
3736ex 450 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → ((𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2) → 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
3837eximdv 1843 . . . . . . . . . 10 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
3938ex 450 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
4039com23 86 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
4115, 40sylbid 230 . . . . . . 7 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑤 ∈ (2 WWalksN 𝐺) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
4241imp 445 . . . . . 6 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑤 ∈ (2 WWalksN 𝐺)) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
4342pm4.71d 665 . . . . 5 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑤 ∈ (2 WWalksN 𝐺)) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ↔ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
4443bicomd 213 . . . 4 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑤 ∈ (2 WWalksN 𝐺)) → ((((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
4544rabbidva 3176 . . 3 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → {𝑤 ∈ (2 WWalksN 𝐺) ∣ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)} = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
4610, 45eqtrd 2655 . 2 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
471iswspthsnon 26617 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
48473ad2ant2 1081 . 2 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
491iswwlksnon 26616 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(2 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
50493ad2ant2 1081 . 2 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
5146, 48, 503eqtr4d 2665 1 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987   ≠ wne 2790  {crab 2911  Vcvv 3186   class class class wbr 4615  ‘cfv 5849  (class class class)co 6607  0cc0 9883  2c2 11017  #chash 13060  Vtxcvtx 25781   UPGraph cupgr 25878   USGraph cusgr 25944  Walkscwlks 26369  WalksOncwlkson 26370  SPathsOncspthson 26487   WWalksN cwwlksn 26594   WWalksNOn cwwlksnon 26595   WSPathsNOn cwwspthsnon 26597 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 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-ac2 9232  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960 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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-map 7807  df-pm 7808  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-card 8712  df-ac 8886  df-cda 8937  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-3 11027  df-n0 11240  df-xnn0 11311  df-z 11325  df-uz 11635  df-fz 12272  df-fzo 12410  df-hash 13061  df-word 13241  df-concat 13243  df-s1 13244  df-s2 13533  df-s3 13534  df-edg 25847  df-uhgr 25856  df-upgr 25880  df-umgr 25881  df-uspgr 25945  df-usgr 25946  df-wlks 26372  df-wlkson 26373  df-trls 26465  df-trlson 26466  df-pths 26488  df-spths 26489  df-pthson 26490  df-spthson 26491  df-wwlks 26598  df-wwlksn 26599  df-wwlksnon 26600  df-wspthsnon 26602 This theorem is referenced by:  usgr2wspthons3  26732  frgr2wsp1  27060
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