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Theorem 2pthon3v 27655
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 26767 . . . . . . . . . 10 (Edg‘𝐺) = ran (iEdg‘𝐺)
31, 2eqtri 2849 . . . . . . . . 9 𝐸 = ran (iEdg‘𝐺)
43eleq2i 2909 . . . . . . . 8 ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ ran (iEdg‘𝐺))
5 2pthon3v.v . . . . . . . . . . 11 𝑉 = (Vtx‘𝐺)
6 eqid 2826 . . . . . . . . . . 11 (iEdg‘𝐺) = (iEdg‘𝐺)
75, 6uhgrf 26780 . . . . . . . . . 10 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
87ffnd 6514 . . . . . . . . 9 (𝐺 ∈ UHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
9 fvelrnb 6725 . . . . . . . . 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, 10syl5bb 284 . . . . . . 7 (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵}))
123eleq2i 2909 . . . . . . . 8 ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ ran (iEdg‘𝐺))
13 fvelrnb 6725 . . . . . . . . 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, 14syl5bb 284 . . . . . . 7 (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}))
1611, 15anbi12d 630 . . . . . 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 3373 . . . 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 290 . . 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 14205 . . . . . . . 8 ⟨“𝑖𝑗”⟩ = (⟨“𝑖”⟩ ++ ⟨“𝑗”⟩)
2221ovexi 7184 . . . . . . 7 ⟨“𝑖𝑗”⟩ ∈ V
23 df-s3 14206 . . . . . . . 8 ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
2423ovexi 7184 . . . . . . 7 ⟨“𝐴𝐵𝐶”⟩ ∈ V
2522, 24pm3.2i 471 . . . . . 6 (⟨“𝑖𝑗”⟩ ∈ V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ V)
26 eqid 2826 . . . . . . . 8 ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩
27 eqid 2826 . . . . . . . 8 ⟨“𝑖𝑗”⟩ = ⟨“𝑖𝑗”⟩
28 simp-4r 780 . . . . . . . 8 (((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → (𝐴𝑉𝐵𝑉𝐶𝑉))
29 3simpb 1143 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐴𝐵𝐵𝐶))
3029ad3antlr 727 . . . . . . . 8 (((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → (𝐴𝐵𝐵𝐶))
31 eqimss2 4028 . . . . . . . . . 10 (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
32 eqimss2 4028 . . . . . . . . . 10 (((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶} → {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗))
3331, 32anim12i 612 . . . . . . . . 9 ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖) ∧ {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗)))
3433adantl 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)))
35 fveqeq2 6678 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ↔ ((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵}))
3635anbi1d 629 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) ↔ (((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})))
37 eqtr2 2847 . . . . . . . . . . . . . 14 ((((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → {𝐴, 𝐵} = {𝐵, 𝐶})
38 3simpa 1142 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝐴𝑉𝐵𝑉))
39 3simpc 1144 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝐵𝑉𝐶𝑉))
40 preq12bg 4783 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
4138, 39, 40syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐵𝑉𝐶𝑉) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
42 eqneqall 3032 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐵 → (𝐴𝐵𝑖𝑗))
4342com12 32 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴𝐵 → (𝐴 = 𝐵𝑖𝑗))
44433ad2ant1 1127 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐴 = 𝐵𝑖𝑗))
4544com12 32 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 = 𝐵 → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
4645adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 = 𝐵𝐵 = 𝐶) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
47 eqneqall 3032 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐶 → (𝐴𝐶𝑖𝑗))
4847com12 32 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴𝐶 → (𝐴 = 𝐶𝑖𝑗))
49483ad2ant2 1128 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐴 = 𝐶𝑖𝑗))
5049com12 32 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 = 𝐶 → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
5150adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 = 𝐶𝐵 = 𝐵) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
5246, 51jaoi 853 . . . . . . . . . . . . . . . . . . 19 (((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵)) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
5341, 52syl6bi 254 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐵𝑉𝐶𝑉) → ({𝐴, 𝐵} = {𝐵, 𝐶} → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗)))
5453com23 86 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐵𝑉𝐶𝑉) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝑖𝑗)))
5554adantl 482 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝑖𝑗)))
5655imp 407 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝑖𝑗))
5756com12 32 . . . . . . . . . . . . . 14 ({𝐴, 𝐵} = {𝐵, 𝐶} → (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → 𝑖𝑗))
5837, 57syl 17 . . . . . . . . . . . . 13 ((((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → 𝑖𝑗))
5936, 58syl6bi 254 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → 𝑖𝑗)))
6059com23 86 . . . . . . . . . . 11 (𝑖 = 𝑗 → (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → 𝑖𝑗)))
61 2a1 28 . . . . . . . . . . 11 (𝑖𝑗 → (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → 𝑖𝑗)))
6260, 61pm2.61ine 3105 . . . . . . . . . 10 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → 𝑖𝑗))
6362adantr 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))
6463imp 407 . . . . . . . 8 (((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → 𝑖𝑗)
65 simplr2 1210 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) → 𝐴𝐶)
6665adantr 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)
6726, 27, 28, 30, 34, 5, 6, 64, 662pthond 27654 . . . . . . 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”⟩)
68 s2len 14246 . . . . . . 7 (♯‘⟨“𝑖𝑗”⟩) = 2
6967, 68jctir 521 . . . . . 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))
70 breq12 5068 . . . . . . . 8 ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → (𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ↔ ⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩))
71 fveqeq2 6678 . . . . . . . . 9 (𝑓 = ⟨“𝑖𝑗”⟩ → ((♯‘𝑓) = 2 ↔ (♯‘⟨“𝑖𝑗”⟩) = 2))
7271adantr 481 . . . . . . . 8 ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((♯‘𝑓) = 2 ↔ (♯‘⟨“𝑖𝑗”⟩) = 2))
7370, 72anbi12d 630 . . . . . . 7 ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2) ↔ (⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘⟨“𝑖𝑗”⟩) = 2)))
7473spc2egv 3604 . . . . . 6 ((⟨“𝑖𝑗”⟩ ∈ V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ V) → ((⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘⟨“𝑖𝑗”⟩) = 2) → ∃𝑓𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2)))
7525, 69, 74mpsyl 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))
7675ex 413 . . . 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)))
7776rexlimdvva 3299 . . 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)))
7820, 77sylbid 241 . 2 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2)))
79783impia 1111 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 207  wa 396  wo 843  w3a 1081   = wceq 1530  wex 1773  wcel 2107  wne 3021  wrex 3144  Vcvv 3500  cdif 3937  wss 3940  c0 4295  𝒫 cpw 4542  {csn 4564  {cpr 4566   class class class wbr 5063  dom cdm 5554  ran crn 5555   Fn wfn 6349  cfv 6354  (class class class)co 7150  2c2 11686  chash 13685   ++ cconcat 13917  ⟨“cs1 13944  ⟨“cs2 14198  ⟨“cs3 14199  Vtxcvtx 26714  iEdgciedg 26715  Edgcedg 26765  UHGraphcuhgr 26774  SPathsOncspthson 27429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ifp 1057  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-map 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12888  df-fzo 13029  df-hash 13686  df-word 13857  df-concat 13918  df-s1 13945  df-s2 14205  df-s3 14206  df-edg 26766  df-uhgr 26776  df-wlks 27314  df-wlkson 27315  df-trls 27407  df-trlson 27408  df-pths 27430  df-spths 27431  df-spthson 27433
This theorem is referenced by:  2pthfrgr  27996
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