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Theorem 2pthon3v 27169
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 26221 . . . . . . . . . 10 (Edg‘𝐺) = ran (iEdg‘𝐺)
31, 2eqtri 2787 . . . . . . . . 9 𝐸 = ran (iEdg‘𝐺)
43eleq2i 2836 . . . . . . . 8 ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ ran (iEdg‘𝐺))
5 2pthon3v.v . . . . . . . . . . 11 𝑉 = (Vtx‘𝐺)
6 eqid 2765 . . . . . . . . . . 11 (iEdg‘𝐺) = (iEdg‘𝐺)
75, 6uhgrf 26237 . . . . . . . . . 10 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
87ffnd 6226 . . . . . . . . 9 (𝐺 ∈ UHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
9 fvelrnb 6434 . . . . . . . . 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 274 . . . . . . 7 (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵}))
123eleq2i 2836 . . . . . . . 8 ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ ran (iEdg‘𝐺))
13 fvelrnb 6434 . . . . . . . . 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 274 . . . . . . 7 (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}))
1611, 15anbi12d 624 . . . . . 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 472 . . . . 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 472 . . . 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 3254 . . . 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 280 . . 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 13880 . . . . . . . 8 ⟨“𝑖𝑗”⟩ = (⟨“𝑖”⟩ ++ ⟨“𝑗”⟩)
2221ovexi 6877 . . . . . . 7 ⟨“𝑖𝑗”⟩ ∈ V
23 df-s3 13881 . . . . . . . 8 ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
2423ovexi 6877 . . . . . . 7 ⟨“𝐴𝐵𝐶”⟩ ∈ V
2522, 24pm3.2i 462 . . . . . 6 (⟨“𝑖𝑗”⟩ ∈ V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ V)
26 eqid 2765 . . . . . . . 8 ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩
27 eqid 2765 . . . . . . . 8 ⟨“𝑖𝑗”⟩ = ⟨“𝑖𝑗”⟩
28 simp-4r 803 . . . . . . . 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 1180 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐴𝐵𝐵𝐶))
3029ad3antlr 722 . . . . . . . 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 3820 . . . . . . . . . 10 (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
32 eqimss2 3820 . . . . . . . . . 10 (((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶} → {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗))
3331, 32anim12i 606 . . . . . . . . 9 ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖) ∧ {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗)))
3433adantl 473 . . . . . . . 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 6386 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ↔ ((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵}))
3635anbi1d 623 . . . . . . . . . . . . 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 2785 . . . . . . . . . . . . . 14 ((((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → {𝐴, 𝐵} = {𝐵, 𝐶})
38 3simpa 1178 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝐴𝑉𝐵𝑉))
39 3simpc 1182 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝐵𝑉𝐶𝑉))
40 preq12bg 4539 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
4138, 39, 40syl2anc 579 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐵𝑉𝐶𝑉) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
42 eqneqall 2948 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐵 → (𝐴𝐵𝑖𝑗))
4342com12 32 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴𝐵 → (𝐴 = 𝐵𝑖𝑗))
44433ad2ant1 1163 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐴 = 𝐵𝑖𝑗))
4544com12 32 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 = 𝐵 → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
4645adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 = 𝐵𝐵 = 𝐶) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
47 eqneqall 2948 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐶 → (𝐴𝐶𝑖𝑗))
4847com12 32 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴𝐶 → (𝐴 = 𝐶𝑖𝑗))
49483ad2ant2 1164 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (𝐴 = 𝐶𝑖𝑗))
5049com12 32 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 = 𝐶 → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
5150adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 = 𝐶𝐵 = 𝐵) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
5246, 51jaoi 883 . . . . . . . . . . . . . . . . . . 19 (((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵)) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗))
5341, 52syl6bi 244 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐵𝑉𝐶𝑉) → ({𝐴, 𝐵} = {𝐵, 𝐶} → ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝑖𝑗)))
5453com23 86 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐵𝑉𝐶𝑉) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝑖𝑗)))
5554adantl 473 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝑖𝑗)))
5655imp 395 . . . . . . . . . . . . . . 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 244 . . . . . . . . . . . 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 3020 . . . . . . . . . 10 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → 𝑖𝑗))
6362adantr 472 . . . . . . . . 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 395 . . . . . . . 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 1277 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) → 𝐴𝐶)
6665adantr 472 . . . . . . . 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 27168 . . . . . . 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 13921 . . . . . . 7 (♯‘⟨“𝑖𝑗”⟩) = 2
6967, 68jctir 516 . . . . . 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 4816 . . . . . . . 8 ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → (𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ↔ ⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩))
71 fveqeq2 6386 . . . . . . . . 9 (𝑓 = ⟨“𝑖𝑗”⟩ → ((♯‘𝑓) = 2 ↔ (♯‘⟨“𝑖𝑗”⟩) = 2))
7271adantr 472 . . . . . . . 8 ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((♯‘𝑓) = 2 ↔ (♯‘⟨“𝑖𝑗”⟩) = 2))
7370, 72anbi12d 624 . . . . . . 7 ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2) ↔ (⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘⟨“𝑖𝑗”⟩) = 2)))
7473spc2egv 3448 . . . . . 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 401 . . . 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 3185 . . 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 231 . 2 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2)))
79783impia 1145 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 197  wa 384  wo 873  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wne 2937  wrex 3056  Vcvv 3350  cdif 3731  wss 3734  c0 4081  𝒫 cpw 4317  {csn 4336  {cpr 4338   class class class wbr 4811  dom cdm 5279  ran crn 5280   Fn wfn 6065  cfv 6070  (class class class)co 6844  2c2 11329  chash 13324   ++ cconcat 13544  ⟨“cs1 13569  ⟨“cs2 13873  ⟨“cs3 13874  Vtxcvtx 26168  iEdgciedg 26169  Edgcedg 26219  UHGraphcuhgr 26231  SPathsOncspthson 26906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ifp 1086  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-oadd 7770  df-er 7949  df-map 8064  df-pm 8065  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-card 9018  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-nn 11277  df-2 11337  df-3 11338  df-n0 11541  df-z 11627  df-uz 11890  df-fz 12537  df-fzo 12677  df-hash 13325  df-word 13490  df-concat 13545  df-s1 13570  df-s2 13880  df-s3 13881  df-edg 26220  df-uhgr 26233  df-wlks 26789  df-wlkson 26790  df-trls 26884  df-trlson 26885  df-pths 26907  df-spths 26908  df-spthson 26910
This theorem is referenced by:  2pthfrgr  27567
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