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Theorem wpthswwlks2onOLD 27125
Description: Obsolete version of wpthswwlks2on 27124 as of 16-Mar-2022. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 13-May-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
wpthswwlks2onOLD.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
wpthswwlks2onOLD ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))

Proof of Theorem wpthswwlks2onOLD
Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wpthswwlks2onOLD.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
21wwlknonOLD 27005 . . . . . . 7 ((𝐴𝑉𝐵𝑉) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
323ad2ant2 1157 . . . . . 6 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
43anbi1d 617 . . . . 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 1109 . . . . . . 7 ((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
65anbi1i 612 . . . . . 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 456 . . . . . 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 266 . . . . 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 278 . . . 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 3387 . . 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 26314 . . . . . . . . . . 11 (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
12 wlklnwwlknupgr 27035 . . . . . . . . . . 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 214 . . . . . . . . 9 (𝐺 ∈ USGraph → (𝑤 ∈ (2 WWalksN 𝐺) ↔ ∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)))
15143ad2ant1 1156 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑤 ∈ (2 WWalksN 𝐺) ↔ ∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)))
16 simprl 778 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝑓(Walks‘𝐺)𝑤)
17 simprl 778 . . . . . . . . . . . . . . 15 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (𝑤‘0) = 𝐴)
1817adantr 468 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘0) = 𝐴)
19 fveq2 6417 . . . . . . . . . . . . . . . 16 ((♯‘𝑓) = 2 → (𝑤‘(♯‘𝑓)) = (𝑤‘2))
2019ad2antll 711 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘(♯‘𝑓)) = (𝑤‘2))
21 simprr 780 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (𝑤‘2) = 𝐵)
2221adantr 468 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘2) = 𝐵)
2320, 22eqtrd 2851 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘(♯‘𝑓)) = 𝐵)
24 simpll2 1264 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝐴𝑉𝐵𝑉))
25 vex 3405 . . . . . . . . . . . . . . . 16 𝑓 ∈ V
26 vex 3405 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
2725, 26pm3.2i 458 . . . . . . . . . . . . . . 15 (𝑓 ∈ V ∧ 𝑤 ∈ V)
281iswlkon 26803 . . . . . . . . . . . . . . 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 576 . . . . . . . . . . . . . 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 1435 . . . . . . . . . . . . 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 1262 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝐺 ∈ USGraph)
32 simprr 780 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (♯‘𝑓) = 2)
33 simpll3 1266 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝐴𝐵)
34 usgr2wlkspth 26905 . . . . . . . . . . . . . 14 ((𝐺 ∈ USGraph ∧ (♯‘𝑓) = 2 ∧ 𝐴𝐵) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
3531, 32, 33, 34syl3anc 1483 . . . . . . . . . . . . 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 223 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
3736ex 399 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
3837eximdv 2008 . . . . . . . . . 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 399 . . . . . . . . 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 231 . . . . . . 7 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑤 ∈ (2 WWalksN 𝐺) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
4241imp 395 . . . . . 6 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑤 ∈ (2 WWalksN 𝐺)) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
4342pm4.71d 553 . . . . 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 214 . . . 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 3389 . . 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 2851 . 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)})
471iswspthsnonOLD 27002 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
48473ad2ant2 1157 . 2 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
491iswwlksnonOLD 26998 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(2 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
50493ad2ant2 1157 . 2 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
5146, 48, 503eqtr4d 2861 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 197  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2157  wne 2989  {crab 3111  Vcvv 3402   class class class wbr 4855  cfv 6110  (class class class)co 6883  0cc0 10230  2c2 11367  chash 13356  Vtxcvtx 26110  UPGraphcupgr 26211  USGraphcusgr 26281  Walkscwlks 26742  WalksOncwlkson 26743  SPathsOncspthson 26861   WWalksN cwwlksn 26969   WWalksNOn cwwlksnon 26970   WSPathsNOn cwwspthsnon 26972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7188  ax-ac2 9579  ax-cnex 10286  ax-resscn 10287  ax-1cn 10288  ax-icn 10289  ax-addcl 10290  ax-addrcl 10291  ax-mulcl 10292  ax-mulrcl 10293  ax-mulcom 10294  ax-addass 10295  ax-mulass 10296  ax-distr 10297  ax-i2m1 10298  ax-1ne0 10299  ax-1rid 10300  ax-rnegex 10301  ax-rrecex 10302  ax-cnre 10303  ax-pre-lttri 10304  ax-pre-lttrn 10305  ax-pre-ltadd 10306  ax-pre-mulgt0 10307
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ifp 1079  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-se 5284  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5906  df-ord 5952  df-on 5953  df-lim 5954  df-suc 5955  df-iota 6073  df-fun 6112  df-fn 6113  df-f 6114  df-f1 6115  df-fo 6116  df-f1o 6117  df-fv 6118  df-isom 6119  df-riota 6844  df-ov 6886  df-oprab 6887  df-mpt2 6888  df-om 7305  df-1st 7407  df-2nd 7408  df-wrecs 7651  df-recs 7713  df-rdg 7751  df-1o 7805  df-2o 7806  df-oadd 7809  df-er 7988  df-map 8103  df-pm 8104  df-en 8202  df-dom 8203  df-sdom 8204  df-fin 8205  df-card 9057  df-ac 9231  df-cda 9284  df-pnf 10370  df-mnf 10371  df-xr 10372  df-ltxr 10373  df-le 10374  df-sub 10562  df-neg 10563  df-nn 11315  df-2 11375  df-3 11376  df-n0 11579  df-xnn0 11649  df-z 11663  df-uz 11924  df-fz 12569  df-fzo 12709  df-hash 13357  df-word 13529  df-concat 13531  df-s1 13532  df-s2 13836  df-s3 13837  df-edg 26176  df-uhgr 26189  df-upgr 26213  df-umgr 26214  df-uspgr 26282  df-usgr 26283  df-wlks 26745  df-wlkson 26746  df-trls 26839  df-trlson 26840  df-pths 26862  df-spths 26863  df-pthson 26864  df-spthson 26865  df-wwlks 26973  df-wwlksn 26974  df-wwlksnon 26975  df-wspthsnon 26977
This theorem is referenced by: (None)
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