Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgreg2spot Structured version   Visualization version   GIF version

Theorem usgreg2spot 26594
 Description: In a finite k-regular graph the set of all paths of length two is the union of all the paths of length 2 over the vertices which are in the middle of such a path. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
Hypothesis
Ref Expression
usgreghash2spot.m 𝑀 = (𝑎𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)})
Assertion
Ref Expression
usgreg2spot ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝑉 2SPathsOt 𝐸) = 𝑥𝑉 (𝑀𝑥)))
Distinct variable groups:   𝑡,𝐸,𝑥,𝑎   𝑉,𝑎,𝑡,𝑥   𝐸,𝑎,𝑣,𝑡   𝑣,𝑉,𝑎   𝑥,𝐾   𝑣,𝑀   𝑥,𝑣
Allowed substitution hints:   𝐾(𝑣,𝑡,𝑎)   𝑀(𝑥,𝑡,𝑎)

Proof of Theorem usgreg2spot
Dummy variables 𝑝 𝑦 𝑧 𝑓 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrav 25867 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2 el2pthsot 26408 . . . . . . . . . 10 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑦𝑉𝑥𝑉𝑧𝑉 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2))))))
31, 2syl 17 . . . . . . . . 9 (𝑉 USGrph 𝐸 → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑦𝑉𝑥𝑉𝑧𝑉 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2))))))
4 simpr 476 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 USGrph 𝐸𝑦𝑉) → 𝑦𝑉)
54ad2antrr 758 . . . . . . . . . . . . . . . . . . 19 ((((𝑉 USGrph 𝐸𝑦𝑉) ∧ 𝑥𝑉) ∧ 𝑧𝑉) → 𝑦𝑉)
6 simpr 476 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 USGrph 𝐸𝑦𝑉) ∧ 𝑥𝑉) → 𝑥𝑉)
76adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝑉 USGrph 𝐸𝑦𝑉) ∧ 𝑥𝑉) ∧ 𝑧𝑉) → 𝑥𝑉)
8 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((((𝑉 USGrph 𝐸𝑦𝑉) ∧ 𝑥𝑉) ∧ 𝑧𝑉) → 𝑧𝑉)
95, 7, 83jca 1235 . . . . . . . . . . . . . . . . . 18 ((((𝑉 USGrph 𝐸𝑦𝑉) ∧ 𝑥𝑉) ∧ 𝑧𝑉) → (𝑦𝑉𝑥𝑉𝑧𝑉))
10 ot2ndg 7074 . . . . . . . . . . . . . . . . . 18 ((𝑦𝑉𝑥𝑉𝑧𝑉) → (2nd ‘(1st ‘⟨𝑦, 𝑥, 𝑧⟩)) = 𝑥)
119, 10syl 17 . . . . . . . . . . . . . . . . 17 ((((𝑉 USGrph 𝐸𝑦𝑉) ∧ 𝑥𝑉) ∧ 𝑧𝑉) → (2nd ‘(1st ‘⟨𝑦, 𝑥, 𝑧⟩)) = 𝑥)
12 eqidd 2611 . . . . . . . . . . . . . . . . . 18 ((((𝑉 USGrph 𝐸𝑦𝑉) ∧ 𝑥𝑉) ∧ 𝑧𝑉) → ⟨𝑦, 𝑥, 𝑧⟩ = ⟨𝑦, 𝑥, 𝑧⟩)
13 otel3xp 5077 . . . . . . . . . . . . . . . . . 18 ((⟨𝑦, 𝑥, 𝑧⟩ = ⟨𝑦, 𝑥, 𝑧⟩ ∧ (𝑦𝑉𝑥𝑉𝑧𝑉)) → ⟨𝑦, 𝑥, 𝑧⟩ ∈ ((𝑉 × 𝑉) × 𝑉))
1412, 9, 13syl2anc 691 . . . . . . . . . . . . . . . . 17 ((((𝑉 USGrph 𝐸𝑦𝑉) ∧ 𝑥𝑉) ∧ 𝑧𝑉) → ⟨𝑦, 𝑥, 𝑧⟩ ∈ ((𝑉 × 𝑉) × 𝑉))
1511, 14jca 553 . . . . . . . . . . . . . . . 16 ((((𝑉 USGrph 𝐸𝑦𝑉) ∧ 𝑥𝑉) ∧ 𝑧𝑉) → ((2nd ‘(1st ‘⟨𝑦, 𝑥, 𝑧⟩)) = 𝑥 ∧ ⟨𝑦, 𝑥, 𝑧⟩ ∈ ((𝑉 × 𝑉) × 𝑉)))
16 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ → (1st𝑝) = (1st ‘⟨𝑦, 𝑥, 𝑧⟩))
1716fveq2d 6107 . . . . . . . . . . . . . . . . . 18 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ → (2nd ‘(1st𝑝)) = (2nd ‘(1st ‘⟨𝑦, 𝑥, 𝑧⟩)))
1817eqeq1d 2612 . . . . . . . . . . . . . . . . 17 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ → ((2nd ‘(1st𝑝)) = 𝑥 ↔ (2nd ‘(1st ‘⟨𝑦, 𝑥, 𝑧⟩)) = 𝑥))
19 eleq1 2676 . . . . . . . . . . . . . . . . 17 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ → (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ↔ ⟨𝑦, 𝑥, 𝑧⟩ ∈ ((𝑉 × 𝑉) × 𝑉)))
2018, 19anbi12d 743 . . . . . . . . . . . . . . . 16 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ → (((2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) ↔ ((2nd ‘(1st ‘⟨𝑦, 𝑥, 𝑧⟩)) = 𝑥 ∧ ⟨𝑦, 𝑥, 𝑧⟩ ∈ ((𝑉 × 𝑉) × 𝑉))))
2115, 20syl5ibr 235 . . . . . . . . . . . . . . 15 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ → ((((𝑉 USGrph 𝐸𝑦𝑉) ∧ 𝑥𝑉) ∧ 𝑧𝑉) → ((2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
2221adantr 480 . . . . . . . . . . . . . 14 ((𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2)))) → ((((𝑉 USGrph 𝐸𝑦𝑉) ∧ 𝑥𝑉) ∧ 𝑧𝑉) → ((2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
2322com12 32 . . . . . . . . . . . . 13 ((((𝑉 USGrph 𝐸𝑦𝑉) ∧ 𝑥𝑉) ∧ 𝑧𝑉) → ((𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2)))) → ((2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
2423rexlimdva 3013 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐸𝑦𝑉) ∧ 𝑥𝑉) → (∃𝑧𝑉 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2)))) → ((2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
2524reximdva 3000 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸𝑦𝑉) → (∃𝑥𝑉𝑧𝑉 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2)))) → ∃𝑥𝑉 ((2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
26 r19.41v 3070 . . . . . . . . . . 11 (∃𝑥𝑉 ((2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) ↔ (∃𝑥𝑉 (2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)))
2725, 26syl6ib 240 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝑦𝑉) → (∃𝑥𝑉𝑧𝑉 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2)))) → (∃𝑥𝑉 (2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
2827rexlimdva 3013 . . . . . . . . 9 (𝑉 USGrph 𝐸 → (∃𝑦𝑉𝑥𝑉𝑧𝑉 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2)))) → (∃𝑥𝑉 (2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
293, 28sylbid 229 . . . . . . . 8 (𝑉 USGrph 𝐸 → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) → (∃𝑥𝑉 (2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
3029ad2antrr 758 . . . . . . 7 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) → (∃𝑥𝑉 (2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
3130pm4.71rd 665 . . . . . 6 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ↔ ((∃𝑥𝑉 (2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸))))
32 anass 679 . . . . . . . . 9 (((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ∧ (2nd ‘(1st𝑝)) = 𝑥) ↔ (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑝)) = 𝑥)))
3332bicomi 213 . . . . . . . 8 ((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑝)) = 𝑥)) ↔ ((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ∧ (2nd ‘(1st𝑝)) = 𝑥))
3433rexbii 3023 . . . . . . 7 (∃𝑥𝑉 (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑝)) = 𝑥)) ↔ ∃𝑥𝑉 ((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ∧ (2nd ‘(1st𝑝)) = 𝑥))
35 ancom 465 . . . . . . . 8 (((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ∧ ∃𝑥𝑉 (2nd ‘(1st𝑝)) = 𝑥) ↔ (∃𝑥𝑉 (2nd ‘(1st𝑝)) = 𝑥 ∧ (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸))))
36 r19.42v 3073 . . . . . . . 8 (∃𝑥𝑉 ((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ∧ (2nd ‘(1st𝑝)) = 𝑥) ↔ ((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ∧ ∃𝑥𝑉 (2nd ‘(1st𝑝)) = 𝑥))
37 anass 679 . . . . . . . 8 (((∃𝑥𝑉 (2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ↔ (∃𝑥𝑉 (2nd ‘(1st𝑝)) = 𝑥 ∧ (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸))))
3835, 36, 373bitr4i 291 . . . . . . 7 (∃𝑥𝑉 ((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ∧ (2nd ‘(1st𝑝)) = 𝑥) ↔ ((∃𝑥𝑉 (2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)))
3934, 38bitri 263 . . . . . 6 (∃𝑥𝑉 (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑝)) = 𝑥)) ↔ ((∃𝑥𝑉 (2nd ‘(1st𝑝)) = 𝑥𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)))
4031, 39syl6bbr 277 . . . . 5 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑥𝑉 (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑝)) = 𝑥))))
41 simpr 476 . . . . . . . . 9 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑥𝑉) → 𝑥𝑉)
42 3xpexg 6859 . . . . . . . . . . 11 (𝑉 ∈ Fin → ((𝑉 × 𝑉) × 𝑉) ∈ V)
43 rabexg 4739 . . . . . . . . . . 11 (((𝑉 × 𝑉) × 𝑉) ∈ V → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑥)} ∈ V)
4442, 43syl 17 . . . . . . . . . 10 (𝑉 ∈ Fin → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑥)} ∈ V)
4544ad3antlr 763 . . . . . . . . 9 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑥𝑉) → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑥)} ∈ V)
46 eqeq2 2621 . . . . . . . . . . . 12 (𝑎 = 𝑥 → ((2nd ‘(1st𝑡)) = 𝑎 ↔ (2nd ‘(1st𝑡)) = 𝑥))
4746anbi2d 736 . . . . . . . . . . 11 (𝑎 = 𝑥 → ((𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎) ↔ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑥)))
4847rabbidv 3164 . . . . . . . . . 10 (𝑎 = 𝑥 → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)} = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑥)})
49 usgreghash2spot.m . . . . . . . . . 10 𝑀 = (𝑎𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)})
5048, 49fvmptg 6189 . . . . . . . . 9 ((𝑥𝑉 ∧ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑥)} ∈ V) → (𝑀𝑥) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑥)})
5141, 45, 50syl2anc 691 . . . . . . . 8 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑥𝑉) → (𝑀𝑥) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑥)})
5251eleq2d 2673 . . . . . . 7 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑥𝑉) → (𝑝 ∈ (𝑀𝑥) ↔ 𝑝 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑥)}))
53 eleq1 2676 . . . . . . . . 9 (𝑡 = 𝑝 → (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ↔ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)))
54 fveq2 6103 . . . . . . . . . . 11 (𝑡 = 𝑝 → (1st𝑡) = (1st𝑝))
5554fveq2d 6107 . . . . . . . . . 10 (𝑡 = 𝑝 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑝)))
5655eqeq1d 2612 . . . . . . . . 9 (𝑡 = 𝑝 → ((2nd ‘(1st𝑡)) = 𝑥 ↔ (2nd ‘(1st𝑝)) = 𝑥))
5753, 56anbi12d 743 . . . . . . . 8 (𝑡 = 𝑝 → ((𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑥) ↔ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑝)) = 𝑥)))
5857elrab 3331 . . . . . . 7 (𝑝 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑥)} ↔ (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑝)) = 𝑥)))
5952, 58syl6rbb 276 . . . . . 6 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑥𝑉) → ((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑝)) = 𝑥)) ↔ 𝑝 ∈ (𝑀𝑥)))
6059rexbidva 3031 . . . . 5 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (∃𝑥𝑉 (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑝)) = 𝑥)) ↔ ∃𝑥𝑉 𝑝 ∈ (𝑀𝑥)))
6140, 60bitrd 267 . . . 4 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑥𝑉 𝑝 ∈ (𝑀𝑥)))
62 eliun 4460 . . . 4 (𝑝 𝑥𝑉 (𝑀𝑥) ↔ ∃𝑥𝑉 𝑝 ∈ (𝑀𝑥))
6361, 62syl6bbr 277 . . 3 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ↔ 𝑝 𝑥𝑉 (𝑀𝑥)))
6463eqrdv 2608 . 2 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑉 2SPathsOt 𝐸) = 𝑥𝑉 (𝑀𝑥))
6564ex 449 1 ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝑉 2SPathsOt 𝐸) = 𝑥𝑉 (𝑀𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ⟨cotp 4133  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Fincfn 7841  0cc0 9815  1c1 9816  2c2 10947  #chash 12979   USGrph cusg 25859   SPaths cspath 26029   2SPathsOt c2spthot 26383   VDeg cvdg 26420 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-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-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-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-ot 4134  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-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-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-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-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-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-wlkon 26042  df-spthon 26045  df-2spthonot 26387  df-2spthsot 26388 This theorem is referenced by:  usgreghash2spot  26596
 Copyright terms: Public domain W3C validator