Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wpthswwlks2on Structured version   Visualization version   GIF version

Theorem wpthswwlks2on 41164
 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 41053 . . . . . . 7 ((𝐴𝑉𝐵𝑉) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑤 ∈ (2 WWalkSN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
323ad2ant2 1076 . . . . . 6 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑤 ∈ (2 WWalkSN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
43anbi1d 737 . . . . 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 1035 . . . . . . 7 ((𝑤 ∈ (2 WWalkSN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ↔ (𝑤 ∈ (2 WWalkSN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
65anbi1i 727 . . . . . 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 679 . . . . . 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 263 . . . . 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 275 . . . 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 3162 . . 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 40412 . . . . . . . . . . 11 (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph )
12 1wlklnwwlknupgr 41083 . . . . . . . . . . 11 (𝐺 ∈ UPGraph → (∃𝑓(𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2) ↔ 𝑤 ∈ (2 WWalkSN 𝐺)))
1311, 12syl 17 . . . . . . . . . 10 (𝐺 ∈ USGraph → (∃𝑓(𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2) ↔ 𝑤 ∈ (2 WWalkSN 𝐺)))
1413bicomd 212 . . . . . . . . 9 (𝐺 ∈ USGraph → (𝑤 ∈ (2 WWalkSN 𝐺) ↔ ∃𝑓(𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)))
15143ad2ant1 1075 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑤 ∈ (2 WWalkSN 𝐺) ↔ ∃𝑓(𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)))
16 simprl 790 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → 𝑓(1Walks‘𝐺)𝑤)
17 simprl 790 . . . . . . . . . . . . . . 15 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (𝑤‘0) = 𝐴)
1817adantr 480 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (𝑤‘0) = 𝐴)
19 fveq2 6103 . . . . . . . . . . . . . . . 16 ((#‘𝑓) = 2 → (𝑤‘(#‘𝑓)) = (𝑤‘2))
2019ad2antll 761 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (𝑤‘(#‘𝑓)) = (𝑤‘2))
21 simprr 792 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (𝑤‘2) = 𝐵)
2221adantr 480 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (𝑤‘2) = 𝐵)
2320, 22eqtrd 2644 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (𝑤‘(#‘𝑓)) = 𝐵)
24 simpll2 1094 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (𝐴𝑉𝐵𝑉))
25 vex 3176 . . . . . . . . . . . . . . . 16 𝑓 ∈ V
26 vex 3176 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
2725, 26pm3.2i 470 . . . . . . . . . . . . . . 15 (𝑓 ∈ V ∧ 𝑤 ∈ V)
281iswlkOn 40865 . . . . . . . . . . . . . . 15 (((𝐴𝑉𝐵𝑉) ∧ (𝑓 ∈ V ∧ 𝑤 ∈ V)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤 ↔ (𝑓(1Walks‘𝐺)𝑤 ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘(#‘𝑓)) = 𝐵)))
2924, 27, 28sylancl 693 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤 ↔ (𝑓(1Walks‘𝐺)𝑤 ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘(#‘𝑓)) = 𝐵)))
3016, 18, 23, 29mpbir3and 1238 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → 𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤)
31 simpll1 1093 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → 𝐺 ∈ USGraph )
32 simprr 792 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (#‘𝑓) = 2)
33 simpll3 1095 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → 𝐴𝐵)
34 usgr2wlkspth 40965 . . . . . . . . . . . . . 14 ((𝐺 ∈ USGraph ∧ (#‘𝑓) = 2 ∧ 𝐴𝐵) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
3531, 32, 33, 34syl3anc 1318 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
3630, 35mpbid 221 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2)) → 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
3736ex 449 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → ((𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2) → 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
3837eximdv 1833 . . . . . . . . . 10 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (∃𝑓(𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
3938ex 449 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → (∃𝑓(𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
4039com23 84 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (∃𝑓(𝑓(1Walks‘𝐺)𝑤 ∧ (#‘𝑓) = 2) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
4115, 40sylbid 229 . . . . . . 7 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑤 ∈ (2 WWalkSN 𝐺) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
4241imp 444 . . . . . 6 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑤 ∈ (2 WWalkSN 𝐺)) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
4342pm4.71d 664 . . . . 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 212 . . . 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 3163 . . 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 2644 . 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 41052 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
48473ad2ant2 1076 . 2 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
491iswwlksnon 41051 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(2 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
50493ad2ant2 1076 . 2 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
5146, 48, 503eqtr4d 2654 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  0cc0 9815  2c2 10947  #chash 12979  Vtxcvtx 25673   UPGraph cupgr 25747   USGraph cusgr 40379  1Walksc1wlks 40796  WalksOncwlkson 40798  SPathsOncspthson 40922   WWalkSN cwwlksn 41029   WWalksNOn cwwlksnon 41030   WSPathsNOn cwwspthsnon 41032 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-ac 8822  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-uhgr 25724  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-1wlks 40800  df-wlks 40801  df-wlkson 40802  df-trls 40901  df-trlson 40902  df-pths 40923  df-spths 40924  df-pthson 40925  df-spthson 40926  df-wwlks 41033  df-wwlksn 41034  df-wwlksnon 41035  df-wspthsnon 41037 This theorem is referenced by:  usgr2wspthons3  41167  frgr2wsp1  41495
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