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Theorem elwspths2spth 27258
Description: A simple path of length 2 between two vertices as length 3 string in a pseudograph. (Contributed by Alexander van der Vekens, 28-Feb-2018.) (Revised by AV, 18-May-2021.) (Proof shortened by AV, 16-Mar-2022.)
Hypothesis
Ref Expression
elwwlks2.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
elwspths2spth (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
Distinct variable groups:   𝐺,𝑎,𝑏,𝑐,𝑓,𝑝   𝑉,𝑎,𝑏,𝑐,𝑓,𝑝   𝑊,𝑎,𝑏,𝑐,𝑓,𝑝

Proof of Theorem elwspths2spth
StepHypRef Expression
1 elwwlks2.v . . . 4 𝑉 = (Vtx‘𝐺)
21wspthsnwspthsnon 27203 . . 3 (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎𝑉𝑐𝑉 𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))
32a1i 11 . 2 (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎𝑉𝑐𝑉 𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)))
41elwspths2on 27250 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉𝑐𝑉) → (𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))))
543expb 1150 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑎𝑉𝑐𝑉)) → (𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))))
652rexbidva 3237 . 2 (𝐺 ∈ UPGraph → (∃𝑎𝑉𝑐𝑉 𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑎𝑉𝑐𝑉𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))))
7 rexcom 3280 . . . 4 (∃𝑐𝑉𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ ∃𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)))
8 wspthnon 27112 . . . . . . 7 (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩))
9 ancom 453 . . . . . . . . 9 ((⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩) ↔ (∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)))
10 19.41v 2045 . . . . . . . . 9 (∃𝑓(𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ (∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)))
119, 10bitr4i 270 . . . . . . . 8 ((⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩) ↔ ∃𝑓(𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)))
12 simpr 478 . . . . . . . . . . . . . 14 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉) → 𝑎𝑉)
13 simpr 478 . . . . . . . . . . . . . 14 ((𝑏𝑉𝑐𝑉) → 𝑐𝑉)
1412, 13anim12i 607 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑎𝑉𝑐𝑉))
15 vex 3388 . . . . . . . . . . . . . 14 𝑓 ∈ V
16 s3cli 13966 . . . . . . . . . . . . . 14 ⟨“𝑎𝑏𝑐”⟩ ∈ Word V
1715, 16pm3.2i 463 . . . . . . . . . . . . 13 (𝑓 ∈ V ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)
181isspthonpth 27003 . . . . . . . . . . . . 13 (((𝑎𝑉𝑐𝑉) ∧ (𝑓 ∈ V ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)) → (𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ↔ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)))
1914, 17, 18sylancl 581 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ↔ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)))
20 wwlknon 27111 . . . . . . . . . . . . 13 (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))
21 2nn0 11599 . . . . . . . . . . . . . . . 16 2 ∈ ℕ0
22 iswwlksn 27089 . . . . . . . . . . . . . . . 16 (2 ∈ ℕ0 → (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1))))
2321, 22ax-mp 5 . . . . . . . . . . . . . . 15 (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)))
2423a1i 11 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1))))
25243anbi1d 1565 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → ((⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) ↔ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
2620, 25syl5bb 275 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
2719, 26anbi12d 625 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → ((𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))))
2827adantr 473 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))))
2916a1i 11 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)
30 simprl1 1282 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩)
31 spthiswlk 26982 . . . . . . . . . . . . . . . . . . . 20 (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ → 𝑓(Walks‘𝐺)⟨“𝑎𝑏𝑐”⟩)
32 wlklenvm1 26871 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walks‘𝐺)⟨“𝑎𝑏𝑐”⟩ → (♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1))
33 simpl 475 . . . . . . . . . . . . . . . . . . . . . 22 (((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1))
34 oveq1 6885 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = ((2 + 1) − 1))
35 2cn 11388 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 ∈ ℂ
36 pncan1 10746 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (2 ∈ ℂ → ((2 + 1) − 1) = 2)
3735, 36ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((2 + 1) − 1) = 2
3834, 37syl6eq 2849 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
3938adantl 474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
40393ad2ant1 1164 . . . . . . . . . . . . . . . . . . . . . . 23 (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
4140adantl 474 . . . . . . . . . . . . . . . . . . . . . 22 (((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
4233, 41eqtrd 2833 . . . . . . . . . . . . . . . . . . . . 21 (((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (♯‘𝑓) = 2)
4342ex 402 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) → (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → (♯‘𝑓) = 2))
4431, 32, 433syl 18 . . . . . . . . . . . . . . . . . . 19 (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ → (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → (♯‘𝑓) = 2))
45443ad2ant1 1164 . . . . . . . . . . . . . . . . . 18 ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) → (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → (♯‘𝑓) = 2))
4645imp 396 . . . . . . . . . . . . . . . . 17 (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (♯‘𝑓) = 2)
4746adantl 474 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → (♯‘𝑓) = 2)
48 vex 3388 . . . . . . . . . . . . . . . . . . . 20 𝑎 ∈ V
49 s3fv0 13976 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎)
5048, 49ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎
5150eqcomi 2808 . . . . . . . . . . . . . . . . . 18 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0)
52 vex 3388 . . . . . . . . . . . . . . . . . . . 20 𝑏 ∈ V
53 s3fv1 13977 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘1) = 𝑏)
5452, 53ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (⟨“𝑎𝑏𝑐”⟩‘1) = 𝑏
5554eqcomi 2808 . . . . . . . . . . . . . . . . . 18 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1)
56 vex 3388 . . . . . . . . . . . . . . . . . . . 20 𝑐 ∈ V
57 s3fv2 13978 . . . . . . . . . . . . . . . . . . . 20 (𝑐 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)
5856, 57ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐
5958eqcomi 2808 . . . . . . . . . . . . . . . . . 18 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)
6051, 55, 593pm3.2i 1439 . . . . . . . . . . . . . . . . 17 (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))
6160a1i 11 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)))
6230, 47, 613jca 1159 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))))
63 breq2 4847 . . . . . . . . . . . . . . . . 17 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑓(SPaths‘𝐺)𝑝𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
64 fveq1 6410 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘0) = (⟨“𝑎𝑏𝑐”⟩‘0))
6564eqeq2d 2809 . . . . . . . . . . . . . . . . . 18 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑎 = (𝑝‘0) ↔ 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0)))
66 fveq1 6410 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘1) = (⟨“𝑎𝑏𝑐”⟩‘1))
6766eqeq2d 2809 . . . . . . . . . . . . . . . . . 18 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑏 = (𝑝‘1) ↔ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1)))
68 fveq1 6410 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘2) = (⟨“𝑎𝑏𝑐”⟩‘2))
6968eqeq2d 2809 . . . . . . . . . . . . . . . . . 18 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑐 = (𝑝‘2) ↔ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)))
7065, 67, 693anbi123d 1561 . . . . . . . . . . . . . . . . 17 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) ↔ (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))))
7163, 703anbi13d 1563 . . . . . . . . . . . . . . . 16 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) ↔ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)))))
7271ad2antlr 719 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) ↔ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)))))
7362, 72mpbird 249 . . . . . . . . . . . . . 14 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))
7473ex 402 . . . . . . . . . . . . 13 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
7529, 74spcimedv 3480 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → ∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
76 spthiswlk 26982 . . . . . . . . . . . . . . . . . . . . 21 (𝑓(SPaths‘𝐺)𝑝𝑓(Walks‘𝐺)𝑝)
77 wlklenvp1 26868 . . . . . . . . . . . . . . . . . . . . 21 (𝑓(Walks‘𝐺)𝑝 → (♯‘𝑝) = ((♯‘𝑓) + 1))
78 oveq1 6885 . . . . . . . . . . . . . . . . . . . . . . . 24 ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = (2 + 1))
79 2p1e3 11462 . . . . . . . . . . . . . . . . . . . . . . . 24 (2 + 1) = 3
8078, 79syl6eq 2849 . . . . . . . . . . . . . . . . . . . . . . 23 ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = 3)
8180eqeq2d 2809 . . . . . . . . . . . . . . . . . . . . . 22 ((♯‘𝑓) = 2 → ((♯‘𝑝) = ((♯‘𝑓) + 1) ↔ (♯‘𝑝) = 3))
8281biimpcd 241 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑝) = ((♯‘𝑓) + 1) → ((♯‘𝑓) = 2 → (♯‘𝑝) = 3))
8376, 77, 823syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝑓(SPaths‘𝐺)𝑝 → ((♯‘𝑓) = 2 → (♯‘𝑝) = 3))
8483imp 396 . . . . . . . . . . . . . . . . . . 19 ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2) → (♯‘𝑝) = 3)
85843adant3 1163 . . . . . . . . . . . . . . . . . 18 ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (♯‘𝑝) = 3)
8685adantl 474 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → (♯‘𝑝) = 3)
87 eqcom 2806 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = (𝑝‘0) ↔ (𝑝‘0) = 𝑎)
88 eqcom 2806 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = (𝑝‘1) ↔ (𝑝‘1) = 𝑏)
89 eqcom 2806 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = (𝑝‘2) ↔ (𝑝‘2) = 𝑐)
9087, 88, 893anbi123i 1195 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) ↔ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
9190biimpi 208 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
92913ad2ant3 1166 . . . . . . . . . . . . . . . . . 18 ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
9392adantl 474 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
9486, 93jca 508 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → ((♯‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐)))
951wlkpwrd 26867 . . . . . . . . . . . . . . . . . . 19 (𝑓(Walks‘𝐺)𝑝𝑝 ∈ Word 𝑉)
9676, 95syl 17 . . . . . . . . . . . . . . . . . 18 (𝑓(SPaths‘𝐺)𝑝𝑝 ∈ Word 𝑉)
97963ad2ant1 1164 . . . . . . . . . . . . . . . . 17 ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → 𝑝 ∈ Word 𝑉)
9812anim1i 609 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑎𝑉 ∧ (𝑏𝑉𝑐𝑉)))
99 3anass 1117 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑉𝑏𝑉𝑐𝑉) ↔ (𝑎𝑉 ∧ (𝑏𝑉𝑐𝑉)))
10098, 99sylibr 226 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑎𝑉𝑏𝑉𝑐𝑉))
101 eqwrds3 14047 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ Word 𝑉 ∧ (𝑎𝑉𝑏𝑉𝑐𝑉)) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((♯‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))))
10297, 100, 101syl2anr 591 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((♯‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))))
10394, 102mpbird 249 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → 𝑝 = ⟨“𝑎𝑏𝑐”⟩)
10463biimpcd 241 . . . . . . . . . . . . . . . . . . . 20 (𝑓(SPaths‘𝐺)𝑝 → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
1051043ad2ant1 1164 . . . . . . . . . . . . . . . . . . 19 ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
106105adantl 474 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
107106imp 396 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩)
10850a1i 11 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎)
109 fveq2 6411 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑓) = 2 → (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = (⟨“𝑎𝑏𝑐”⟩‘2))
110109, 58syl6eq 2849 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑓) = 2 → (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)
1111103ad2ant2 1165 . . . . . . . . . . . . . . . . . 18 ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)
112111ad2antlr 719 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)
113107, 108, 1123jca 1159 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐))
114 wlkiswwlks1 27124 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐺 ∈ UPGraph → (𝑓(Walks‘𝐺)𝑝𝑝 ∈ (WWalks‘𝐺)))
115114adantr 473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉) → (𝑓(Walks‘𝐺)𝑝𝑝 ∈ (WWalks‘𝐺)))
116115adantr 473 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑓(Walks‘𝐺)𝑝𝑝 ∈ (WWalks‘𝐺)))
11776, 116syl5com 31 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓(SPaths‘𝐺)𝑝 → (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → 𝑝 ∈ (WWalks‘𝐺)))
1181173ad2ant1 1164 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → 𝑝 ∈ (WWalks‘𝐺)))
119118impcom 397 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → 𝑝 ∈ (WWalks‘𝐺))
120119adantr 473 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑝 ∈ (WWalks‘𝐺))
121 eleq1 2866 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝 ∈ (WWalks‘𝐺) ↔ ⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺)))
122121bicomd 215 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ↔ 𝑝 ∈ (WWalks‘𝐺)))
123122adantl 474 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ↔ 𝑝 ∈ (WWalks‘𝐺)))
124120, 123mpbird 249 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → ⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺))
125 s3len 13979 . . . . . . . . . . . . . . . . . . 19 (♯‘⟨“𝑎𝑏𝑐”⟩) = 3
126 df-3 11377 . . . . . . . . . . . . . . . . . . 19 3 = (2 + 1)
127125, 126eqtri 2821 . . . . . . . . . . . . . . . . . 18 (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)
128124, 127jctir 517 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)))
12958a1i 11 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)
130128, 108, 1293jca 1159 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))
131113, 130jca 508 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
132103, 131mpdan 679 . . . . . . . . . . . . . 14 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
133132ex 402 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))))
134133exlimdv 2029 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))))
13575, 134impbid 204 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) ↔ ∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
136135adantr 473 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) ↔ ∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
13728, 136bitrd 271 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ ∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
138137exbidv 2017 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (∃𝑓(𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
13911, 138syl5bb 275 . . . . . . 7 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩) ↔ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
1408, 139syl5bb 275 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
141140pm5.32da 575 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → ((𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
1421412rexbidva 3237 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉) → (∃𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ ∃𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
1437, 142syl5bb 275 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉) → (∃𝑐𝑉𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ ∃𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
144143rexbidva 3230 . 2 (𝐺 ∈ UPGraph → (∃𝑎𝑉𝑐𝑉𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
1453, 6, 1443bitrd 297 1 (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wex 1875  wcel 2157  wrex 3090  Vcvv 3385   class class class wbr 4843  cfv 6101  (class class class)co 6878  cc 10222  0cc0 10224  1c1 10225   + caddc 10227  cmin 10556  2c2 11368  3c3 11369  0cn0 11580  chash 13370  Word cword 13534  ⟨“cs3 13927  Vtxcvtx 26231  UPGraphcupgr 26315  Walkscwlks 26846  SPathscspths 26967  SPathsOncspthson 26969  WWalkscwwlks 27076   WWalksN cwwlksn 27077   WWalksNOn cwwlksnon 27078   WSPathsN cwwspthsn 27079   WSPathsNOn cwwspthsnon 27080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-ac2 9573  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ifp 1087  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-se 5272  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-map 8097  df-pm 8098  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-ac 9225  df-cda 9278  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-3 11377  df-n0 11581  df-xnn0 11653  df-z 11667  df-uz 11931  df-fz 12581  df-fzo 12721  df-hash 13371  df-word 13535  df-concat 13591  df-s1 13616  df-s2 13933  df-s3 13934  df-edg 26283  df-uhgr 26293  df-upgr 26317  df-wlks 26849  df-wlkson 26850  df-trls 26945  df-trlson 26946  df-pths 26970  df-spths 26971  df-spthson 26973  df-wwlks 27081  df-wwlksn 27082  df-wwlksnon 27083  df-wspthsn 27084  df-wspthsnon 27085
This theorem is referenced by: (None)
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