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Theorem wpthswwlks2onOLD 27083
Description: Obsolete version of wpthswwlks2on 27082 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 26965 . . . . . . 7 ((𝐴𝑉𝐵𝑉) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
323ad2ant2 1129 . . . . . 6 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
43anbi1d 743 . . . . 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 1081 . . . . . . 7 ((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
65anbi1i 733 . . . . . 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 684 . . . . . 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 3326 . . 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 26276 . . . . . . . . . . 11 (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
12 wlklnwwlknupgr 26995 . . . . . . . . . . 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 1128 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑤 ∈ (2 WWalksN 𝐺) ↔ ∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)))
16 simprl 811 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝑓(Walks‘𝐺)𝑤)
17 simprl 811 . . . . . . . . . . . . . . 15 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (𝑤‘0) = 𝐴)
1817adantr 472 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘0) = 𝐴)
19 fveq2 6352 . . . . . . . . . . . . . . . 16 ((♯‘𝑓) = 2 → (𝑤‘(♯‘𝑓)) = (𝑤‘2))
2019ad2antll 767 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘(♯‘𝑓)) = (𝑤‘2))
21 simprr 813 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (𝑤‘2) = 𝐵)
2221adantr 472 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘2) = 𝐵)
2320, 22eqtrd 2794 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘(♯‘𝑓)) = 𝐵)
24 simpll2 1257 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝐴𝑉𝐵𝑉))
25 vex 3343 . . . . . . . . . . . . . . . 16 𝑓 ∈ V
26 vex 3343 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
2725, 26pm3.2i 470 . . . . . . . . . . . . . . 15 (𝑓 ∈ V ∧ 𝑤 ∈ V)
281iswlkon 26763 . . . . . . . . . . . . . . 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 697 . . . . . . . . . . . . . 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 1428 . . . . . . . . . . . . 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 1255 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝐺 ∈ USGraph)
32 simprr 813 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (♯‘𝑓) = 2)
33 simpll3 1259 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝐴𝐵)
34 usgr2wlkspth 26865 . . . . . . . . . . . . . 14 ((𝐺 ∈ USGraph ∧ (♯‘𝑓) = 2 ∧ 𝐴𝐵) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
3531, 32, 33, 34syl3anc 1477 . . . . . . . . . . . . 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 449 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
3837eximdv 1995 . . . . . . . . . 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 449 . . . . . . . . 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 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 669 . . . . 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 3328 . . 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 2794 . 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 26962 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
48473ad2ant2 1129 . 2 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
491iswwlksnonOLD 26958 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(2 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
50493ad2ant2 1129 . 2 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
5146, 48, 503eqtr4d 2804 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 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  wne 2932  {crab 3054  Vcvv 3340   class class class wbr 4804  cfv 6049  (class class class)co 6813  0cc0 10128  2c2 11262  chash 13311  Vtxcvtx 26073  UPGraphcupgr 26174  USGraphcusgr 26243  Walkscwlks 26702  WalksOncwlkson 26703  SPathsOncspthson 26821   WWalksN cwwlksn 26929   WWalksNOn cwwlksnon 26930   WSPathsNOn cwwspthsnon 26932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-ac2 9477  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-ac 9129  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-xnn0 11556  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-hash 13312  df-word 13485  df-concat 13487  df-s1 13488  df-s2 13793  df-s3 13794  df-edg 26139  df-uhgr 26152  df-upgr 26176  df-umgr 26177  df-uspgr 26244  df-usgr 26245  df-wlks 26705  df-wlkson 26706  df-trls 26799  df-trlson 26800  df-pths 26822  df-spths 26823  df-pthson 26824  df-spthson 26825  df-wwlks 26933  df-wwlksn 26934  df-wwlksnon 26935  df-wspthsnon 26937
This theorem is referenced by: (None)
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