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Theorem 2pthon3v 26708
Description: For a vertex adjacent to two other vertices there is a simple path of length 2 between these other vertices in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 24-Jan-2021.)
Hypotheses
Ref Expression
2pthon3v.v 𝑉 = (Vtx‘𝐺)
2pthon3v.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
2pthon3v (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) → ∃𝑓𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (#‘𝑓) = 2))
Distinct variable groups:   𝐴,𝑓,𝑝   𝐵,𝑓,𝑝   𝐶,𝑓,𝑝   𝑓,𝐺,𝑝
Allowed substitution hints:   𝐸(𝑓,𝑝)   𝑉(𝑓,𝑝)

Proof of Theorem 2pthon3v
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2pthon3v.e . . . . . . . . . 10 𝐸 = (Edg‘𝐺)
2 edgval 25841 . . . . . . . . . 10 (𝐺 ∈ UHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
31, 2syl5eq 2667 . . . . . . . . 9 (𝐺 ∈ UHGraph → 𝐸 = ran (iEdg‘𝐺))
43eleq2d 2684 . . . . . . . 8 (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ ran (iEdg‘𝐺)))
5 2pthon3v.v . . . . . . . . . . 11 𝑉 = (Vtx‘𝐺)
6 eqid 2621 . . . . . . . . . . 11 (iEdg‘𝐺) = (iEdg‘𝐺)
75, 6uhgrf 25853 . . . . . . . . . 10 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
87ffnd 6003 . . . . . . . . 9 (𝐺 ∈ UHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
9 fvelrnb 6200 . . . . . . . . 9 ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → ({𝐴, 𝐵} ∈ ran (iEdg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵}))
108, 9syl 17 . . . . . . . 8 (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ ran (iEdg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵}))
114, 10bitrd 268 . . . . . . 7 (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵}))
123eleq2d 2684 . . . . . . . 8 (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ ran (iEdg‘𝐺)))
13 fvelrnb 6200 . . . . . . . . 9 ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → ({𝐵, 𝐶} ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}))
148, 13syl 17 . . . . . . . 8 (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}))
1512, 14bitrd 268 . . . . . . 7 (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}))
1611, 15anbi12d 746 . . . . . 6 (𝐺 ∈ UHGraph → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})))
1716adantr 481 . . . . 5 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})))
1817adantr 481 . . . 4 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})))
19 reeanv 3097 . . . 4 (∃𝑖 ∈ dom (iEdg‘𝐺)∃𝑗 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) ↔ (∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}))
2018, 19syl6bbr 278 . . 3 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)∃𝑗 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})))
21 df-s2 13530 . . . . . . . 8 ⟨“𝑖𝑗”⟩ = (⟨“𝑖”⟩ ++ ⟨“𝑗”⟩)
22 ovex 6632 . . . . . . . 8 (⟨“𝑖”⟩ ++ ⟨“𝑗”⟩) ∈ V
2321, 22eqeltri 2694 . . . . . . 7 ⟨“𝑖𝑗”⟩ ∈ V
24 df-s3 13531 . . . . . . . 8 ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
25 ovex 6632 . . . . . . . 8 (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩) ∈ V
2624, 25eqeltri 2694 . . . . . . 7 ⟨“𝐴𝐵𝐶”⟩ ∈ V
2723, 26pm3.2i 471 . . . . . 6 (⟨“𝑖𝑗”⟩ ∈ V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ V)
28 eqid 2621 . . . . . . . 8 ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩
29 eqid 2621 . . . . . . . 8 ⟨“𝑖𝑗”⟩ = ⟨“𝑖𝑗”⟩
30 simp-4r 806 . . . . . . . 8 (((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → (𝐴𝑉𝐵𝑉𝐶𝑉))
31 3simpb 1057 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐴𝐵𝐵𝐶))
3231ad3antlr 766 . . . . . . . 8 (((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → (𝐴𝐵𝐵𝐶))
33 eqimss2 3637 . . . . . . . . . 10 (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
34 eqimss2 3637 . . . . . . . . . 10 (((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶} → {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗))
3533, 34anim12i 589 . . . . . . . . 9 ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖) ∧ {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗)))
3635adantl 482 . . . . . . . 8 (((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → ({𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖) ∧ {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗)))
37 fveq2 6148 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘𝑗))
3837eqeq1d 2623 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ↔ ((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵}))
3938anbi1d 740 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) ↔ (((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})))
40 eqtr2 2641 . . . . . . . . . . . . . 14 ((((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → {𝐴, 𝐵} = {𝐵, 𝐶})
41 3simpa 1056 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝐴𝑉𝐵𝑉))
42 3simpc 1058 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝐵𝑉𝐶𝑉))
43 preq12bg 4354 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
4441, 42, 43syl2anc 692 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐵𝑉𝐶𝑉) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
45 eqneqall 2801 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐵 → (𝐴𝐵𝑖𝑗))
4645com12 32 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴𝐵 → (𝐴 = 𝐵𝑖𝑗))
47463ad2ant1 1080 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐴 = 𝐵𝑖𝑗))
4847com12 32 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 = 𝐵 → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
4948adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 = 𝐵𝐵 = 𝐶) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
50 eqneqall 2801 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐶 → (𝐴𝐶𝑖𝑗))
5150com12 32 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴𝐶 → (𝐴 = 𝐶𝑖𝑗))
52513ad2ant2 1081 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐴 = 𝐶𝑖𝑗))
5352com12 32 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 = 𝐶 → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
5453adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 = 𝐶𝐵 = 𝐵) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
5549, 54jaoi 394 . . . . . . . . . . . . . . . . . . 19 (((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵)) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
5644, 55syl6bi 243 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐵𝑉𝐶𝑉) → ({𝐴, 𝐵} = {𝐵, 𝐶} → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗)))
5756com23 86 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐵𝑉𝐶𝑉) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝑖𝑗)))
5857adantl 482 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝑖𝑗)))
5958imp 445 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝑖𝑗))
6059com12 32 . . . . . . . . . . . . . 14 ({𝐴, 𝐵} = {𝐵, 𝐶} → (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → 𝑖𝑗))
6140, 60syl 17 . . . . . . . . . . . . 13 ((((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → 𝑖𝑗))
6239, 61syl6bi 243 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → 𝑖𝑗)))
6362com23 86 . . . . . . . . . . 11 (𝑖 = 𝑗 → (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → 𝑖𝑗)))
64 2a1 28 . . . . . . . . . . 11 (𝑖𝑗 → (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → 𝑖𝑗)))
6563, 64pm2.61ine 2873 . . . . . . . . . 10 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → 𝑖𝑗))
6665adantr 481 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → 𝑖𝑗))
6766imp 445 . . . . . . . 8 (((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → 𝑖𝑗)
68 simplr2 1102 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) → 𝐴𝐶)
6968adantr 481 . . . . . . . 8 (((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → 𝐴𝐶)
7028, 29, 30, 32, 36, 5, 6, 67, 692pthond 26707 . . . . . . 7 (((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → ⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩)
71 s2len 13570 . . . . . . 7 (#‘⟨“𝑖𝑗”⟩) = 2
7270, 71jctir 560 . . . . . 6 (((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → (⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩ ∧ (#‘⟨“𝑖𝑗”⟩) = 2))
73 breq12 4618 . . . . . . . 8 ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → (𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ↔ ⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩))
74 fveq2 6148 . . . . . . . . . 10 (𝑓 = ⟨“𝑖𝑗”⟩ → (#‘𝑓) = (#‘⟨“𝑖𝑗”⟩))
7574eqeq1d 2623 . . . . . . . . 9 (𝑓 = ⟨“𝑖𝑗”⟩ → ((#‘𝑓) = 2 ↔ (#‘⟨“𝑖𝑗”⟩) = 2))
7675adantr 481 . . . . . . . 8 ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((#‘𝑓) = 2 ↔ (#‘⟨“𝑖𝑗”⟩) = 2))
7773, 76anbi12d 746 . . . . . . 7 ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (#‘𝑓) = 2) ↔ (⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩ ∧ (#‘⟨“𝑖𝑗”⟩) = 2)))
7877spc2egv 3281 . . . . . 6 ((⟨“𝑖𝑗”⟩ ∈ V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ V) → ((⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩ ∧ (#‘⟨“𝑖𝑗”⟩) = 2) → ∃𝑓𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (#‘𝑓) = 2)))
7927, 72, 78mpsyl 68 . . . . 5 (((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → ∃𝑓𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (#‘𝑓) = 2))
8079ex 450 . . . 4 ((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → ∃𝑓𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (#‘𝑓) = 2)))
8180rexlimdvva 3031 . . 3 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∃𝑖 ∈ dom (iEdg‘𝐺)∃𝑗 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → ∃𝑓𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (#‘𝑓) = 2)))
8220, 81sylbid 230 . 2 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (#‘𝑓) = 2)))
83823impia 1258 1 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) → ∃𝑓𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (#‘𝑓) = 2))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2908  Vcvv 3186  cdif 3552  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148  {cpr 4150   class class class wbr 4613  dom cdm 5074  ran crn 5075   Fn wfn 5842  cfv 5847  (class class class)co 6604  2c2 11014  #chash 13057   ++ cconcat 13232  ⟨“cs1 13233  ⟨“cs2 13523  ⟨“cs3 13524  Vtxcvtx 25774  iEdgciedg 25775  Edgcedg 25839   UHGraph cuhgr 25847  SPathsOncspthson 26480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-s2 13530  df-s3 13531  df-edg 25840  df-uhgr 25849  df-wlks 26365  df-wlkson 26366  df-trls 26458  df-trlson 26459  df-pths 26481  df-spths 26482  df-spthson 26484
This theorem is referenced by:  2pthfrgr  27012
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