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Theorem uhgr3cyclex 30034
Description: If there are three different vertices in a hypergraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.)
Hypotheses
Ref Expression
uhgr3cyclex.v 𝑉 = (Vtxβ€˜πΊ)
uhgr3cyclex.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
uhgr3cyclex ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢)) ∧ ({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸 ∧ {𝐢, 𝐴} ∈ 𝐸)) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴))
Distinct variable groups:   𝐴,𝑓,𝑝   𝐡,𝑓,𝑝   𝐢,𝑓,𝑝   𝑓,𝐺,𝑝
Allowed substitution hints:   𝐸(𝑓,𝑝)   𝑉(𝑓,𝑝)

Proof of Theorem uhgr3cyclex
Dummy variables 𝑖 𝑗 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uhgr3cyclex.e . . . . . . 7 𝐸 = (Edgβ€˜πΊ)
21eleq2i 2817 . . . . . 6 ({𝐴, 𝐡} ∈ 𝐸 ↔ {𝐴, 𝐡} ∈ (Edgβ€˜πΊ))
3 eqid 2725 . . . . . . 7 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
43uhgredgiedgb 28981 . . . . . 6 (𝐺 ∈ UHGraph β†’ ({𝐴, 𝐡} ∈ (Edgβ€˜πΊ) ↔ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))
52, 4bitrid 282 . . . . 5 (𝐺 ∈ UHGraph β†’ ({𝐴, 𝐡} ∈ 𝐸 ↔ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))
61eleq2i 2817 . . . . . 6 ({𝐡, 𝐢} ∈ 𝐸 ↔ {𝐡, 𝐢} ∈ (Edgβ€˜πΊ))
73uhgredgiedgb 28981 . . . . . 6 (𝐺 ∈ UHGraph β†’ ({𝐡, 𝐢} ∈ (Edgβ€˜πΊ) ↔ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)))
86, 7bitrid 282 . . . . 5 (𝐺 ∈ UHGraph β†’ ({𝐡, 𝐢} ∈ 𝐸 ↔ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)))
91eleq2i 2817 . . . . . 6 ({𝐢, 𝐴} ∈ 𝐸 ↔ {𝐢, 𝐴} ∈ (Edgβ€˜πΊ))
103uhgredgiedgb 28981 . . . . . 6 (𝐺 ∈ UHGraph β†’ ({𝐢, 𝐴} ∈ (Edgβ€˜πΊ) ↔ βˆƒπ‘˜ ∈ dom (iEdgβ€˜πΊ){𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)))
119, 10bitrid 282 . . . . 5 (𝐺 ∈ UHGraph β†’ ({𝐢, 𝐴} ∈ 𝐸 ↔ βˆƒπ‘˜ ∈ dom (iEdgβ€˜πΊ){𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)))
125, 8, 113anbi123d 1432 . . . 4 (𝐺 ∈ UHGraph β†’ (({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸 ∧ {𝐢, 𝐴} ∈ 𝐸) ↔ (βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–) ∧ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ βˆƒπ‘˜ ∈ dom (iEdgβ€˜πΊ){𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜))))
1312adantr 479 . . 3 ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ (({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸 ∧ {𝐢, 𝐴} ∈ 𝐸) ↔ (βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–) ∧ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ βˆƒπ‘˜ ∈ dom (iEdgβ€˜πΊ){𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜))))
14 eqid 2725 . . . . . . . . . . . . . 14 βŸ¨β€œπ΄π΅πΆπ΄β€βŸ© = βŸ¨β€œπ΄π΅πΆπ΄β€βŸ©
15 eqid 2725 . . . . . . . . . . . . . 14 βŸ¨β€œπ‘–π‘—π‘˜β€βŸ© = βŸ¨β€œπ‘–π‘—π‘˜β€βŸ©
16 3simpa 1145 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉))
17 pm3.22 458 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (𝐢 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
18173adant2 1128 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (𝐢 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
1916, 18jca 510 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐢 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
2019adantr 479 . . . . . . . . . . . . . . 15 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢)) β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐢 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
2120ad2antlr 725 . . . . . . . . . . . . . 14 (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) ∧ ((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))) β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐢 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
22 3simpa 1145 . . . . . . . . . . . . . . . . 17 ((𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢) β†’ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢))
23 necom 2984 . . . . . . . . . . . . . . . . . . . 20 (𝐴 β‰  𝐡 ↔ 𝐡 β‰  𝐴)
2423biimpi 215 . . . . . . . . . . . . . . . . . . 19 (𝐴 β‰  𝐡 β†’ 𝐡 β‰  𝐴)
2524anim1ci 614 . . . . . . . . . . . . . . . . . 18 ((𝐴 β‰  𝐡 ∧ 𝐡 β‰  𝐢) β†’ (𝐡 β‰  𝐢 ∧ 𝐡 β‰  𝐴))
26253adant2 1128 . . . . . . . . . . . . . . . . 17 ((𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢) β†’ (𝐡 β‰  𝐢 ∧ 𝐡 β‰  𝐴))
27 necom 2984 . . . . . . . . . . . . . . . . . . 19 (𝐴 β‰  𝐢 ↔ 𝐢 β‰  𝐴)
2827biimpi 215 . . . . . . . . . . . . . . . . . 18 (𝐴 β‰  𝐢 β†’ 𝐢 β‰  𝐴)
29283ad2ant2 1131 . . . . . . . . . . . . . . . . 17 ((𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢) β†’ 𝐢 β‰  𝐴)
3022, 26, 293jca 1125 . . . . . . . . . . . . . . . 16 ((𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢) β†’ ((𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢) ∧ (𝐡 β‰  𝐢 ∧ 𝐡 β‰  𝐴) ∧ 𝐢 β‰  𝐴))
3130adantl 480 . . . . . . . . . . . . . . 15 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢)) β†’ ((𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢) ∧ (𝐡 β‰  𝐢 ∧ 𝐡 β‰  𝐴) ∧ 𝐢 β‰  𝐴))
3231ad2antlr 725 . . . . . . . . . . . . . 14 (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) ∧ ((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))) β†’ ((𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢) ∧ (𝐡 β‰  𝐢 ∧ 𝐡 β‰  𝐴) ∧ 𝐢 β‰  𝐴))
33 eqimss 4031 . . . . . . . . . . . . . . . . . 18 ({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–) β†’ {𝐴, 𝐡} βŠ† ((iEdgβ€˜πΊ)β€˜π‘–))
3433adantl 480 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ {𝐴, 𝐡} βŠ† ((iEdgβ€˜πΊ)β€˜π‘–))
35343ad2ant3 1132 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ {𝐴, 𝐡} βŠ† ((iEdgβ€˜πΊ)β€˜π‘–))
36 eqimss 4031 . . . . . . . . . . . . . . . . . 18 ({𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—) β†’ {𝐡, 𝐢} βŠ† ((iEdgβ€˜πΊ)β€˜π‘—))
3736adantl 480 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) β†’ {𝐡, 𝐢} βŠ† ((iEdgβ€˜πΊ)β€˜π‘—))
38373ad2ant1 1130 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ {𝐡, 𝐢} βŠ† ((iEdgβ€˜πΊ)β€˜π‘—))
39 eqimss 4031 . . . . . . . . . . . . . . . . . 18 ({𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜) β†’ {𝐢, 𝐴} βŠ† ((iEdgβ€˜πΊ)β€˜π‘˜))
4039adantl 480 . . . . . . . . . . . . . . . . 17 ((π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) β†’ {𝐢, 𝐴} βŠ† ((iEdgβ€˜πΊ)β€˜π‘˜))
41403ad2ant2 1131 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ {𝐢, 𝐴} βŠ† ((iEdgβ€˜πΊ)β€˜π‘˜))
4235, 38, 413jca 1125 . . . . . . . . . . . . . . 15 (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ ({𝐴, 𝐡} βŠ† ((iEdgβ€˜πΊ)β€˜π‘–) ∧ {𝐡, 𝐢} βŠ† ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐢, 𝐴} βŠ† ((iEdgβ€˜πΊ)β€˜π‘˜)))
4342adantl 480 . . . . . . . . . . . . . 14 (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) ∧ ((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))) β†’ ({𝐴, 𝐡} βŠ† ((iEdgβ€˜πΊ)β€˜π‘–) ∧ {𝐡, 𝐢} βŠ† ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐢, 𝐴} βŠ† ((iEdgβ€˜πΊ)β€˜π‘˜)))
44 uhgr3cyclex.v . . . . . . . . . . . . . 14 𝑉 = (Vtxβ€˜πΊ)
45 simp3 1135 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ 𝐢 ∈ 𝑉)
46 simp1 1133 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ 𝐴 ∈ 𝑉)
4745, 46jca 510 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (𝐢 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
4847, 29anim12i 611 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢)) β†’ ((𝐢 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ 𝐢 β‰  𝐴))
4948adantl 480 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ ((𝐢 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ 𝐢 β‰  𝐴))
50 pm3.22 458 . . . . . . . . . . . . . . . . 17 (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)) ∧ (𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—))))
51503adant2 1128 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)) ∧ (𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—))))
5244, 1, 3uhgr3cyclexlem 30033 . . . . . . . . . . . . . . . 16 ((((𝐢 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ 𝐢 β‰  𝐴) ∧ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)) ∧ (𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)))) β†’ 𝑖 β‰  𝑗)
5349, 51, 52syl2an 594 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) ∧ ((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))) β†’ 𝑖 β‰  𝑗)
54 3simpc 1147 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉))
55 simp3 1135 . . . . . . . . . . . . . . . . . 18 ((𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢) β†’ 𝐡 β‰  𝐢)
5654, 55anim12i 611 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢)) β†’ ((𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ 𝐡 β‰  𝐢))
5756adantl 480 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ ((𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ 𝐡 β‰  𝐢))
58 3simpc 1147 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ ((π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–))))
5944, 1, 3uhgr3cyclexlem 30033 . . . . . . . . . . . . . . . . 17 ((((𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ 𝐡 β‰  𝐢) ∧ ((π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))) β†’ π‘˜ β‰  𝑖)
6059necomd 2986 . . . . . . . . . . . . . . . 16 ((((𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ 𝐡 β‰  𝐢) ∧ ((π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))) β†’ 𝑖 β‰  π‘˜)
6157, 58, 60syl2an 594 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) ∧ ((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))) β†’ 𝑖 β‰  π‘˜)
6244, 1, 3uhgr3cyclexlem 30033 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ 𝐴 β‰  𝐡) ∧ ((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)))) β†’ 𝑗 β‰  π‘˜)
6362exp31 418 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴 β‰  𝐡 β†’ (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜))) β†’ 𝑗 β‰  π‘˜)))
64633adant3 1129 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (𝐴 β‰  𝐡 β†’ (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜))) β†’ 𝑗 β‰  π‘˜)))
6564com12 32 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 β‰  𝐡 β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜))) β†’ 𝑗 β‰  π‘˜)))
66653ad2ant1 1130 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢) β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜))) β†’ 𝑗 β‰  π‘˜)))
6766impcom 406 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢)) β†’ (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜))) β†’ 𝑗 β‰  π‘˜))
6867adantl 480 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜))) β†’ 𝑗 β‰  π‘˜))
6968com12 32 . . . . . . . . . . . . . . . . 17 (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜))) β†’ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ 𝑗 β‰  π‘˜))
70693adant3 1129 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ 𝑗 β‰  π‘˜))
7170impcom 406 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) ∧ ((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))) β†’ 𝑗 β‰  π‘˜)
7253, 61, 713jca 1125 . . . . . . . . . . . . . 14 (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) ∧ ((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))) β†’ (𝑖 β‰  𝑗 ∧ 𝑖 β‰  π‘˜ ∧ 𝑗 β‰  π‘˜))
73 eqidd 2726 . . . . . . . . . . . . . 14 (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) ∧ ((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))) β†’ 𝐴 = 𝐴)
7414, 15, 21, 32, 43, 44, 3, 72, 733cyclpd 30031 . . . . . . . . . . . . 13 (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) ∧ ((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))) β†’ (βŸ¨β€œπ‘–π‘—π‘˜β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΅πΆπ΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘–π‘—π‘˜β€βŸ©) = 3 ∧ (βŸ¨β€œπ΄π΅πΆπ΄β€βŸ©β€˜0) = 𝐴))
75 s3cli 14862 . . . . . . . . . . . . . . 15 βŸ¨β€œπ‘–π‘—π‘˜β€βŸ© ∈ Word V
7675elexi 3484 . . . . . . . . . . . . . 14 βŸ¨β€œπ‘–π‘—π‘˜β€βŸ© ∈ V
77 s4cli 14863 . . . . . . . . . . . . . . 15 βŸ¨β€œπ΄π΅πΆπ΄β€βŸ© ∈ Word V
7877elexi 3484 . . . . . . . . . . . . . 14 βŸ¨β€œπ΄π΅πΆπ΄β€βŸ© ∈ V
79 breq12 5148 . . . . . . . . . . . . . . 15 ((𝑓 = βŸ¨β€œπ‘–π‘—π‘˜β€βŸ© ∧ 𝑝 = βŸ¨β€œπ΄π΅πΆπ΄β€βŸ©) β†’ (𝑓(Cyclesβ€˜πΊ)𝑝 ↔ βŸ¨β€œπ‘–π‘—π‘˜β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΅πΆπ΄β€βŸ©))
80 fveqeq2 6900 . . . . . . . . . . . . . . . 16 (𝑓 = βŸ¨β€œπ‘–π‘—π‘˜β€βŸ© β†’ ((β™―β€˜π‘“) = 3 ↔ (β™―β€˜βŸ¨β€œπ‘–π‘—π‘˜β€βŸ©) = 3))
8180adantr 479 . . . . . . . . . . . . . . 15 ((𝑓 = βŸ¨β€œπ‘–π‘—π‘˜β€βŸ© ∧ 𝑝 = βŸ¨β€œπ΄π΅πΆπ΄β€βŸ©) β†’ ((β™―β€˜π‘“) = 3 ↔ (β™―β€˜βŸ¨β€œπ‘–π‘—π‘˜β€βŸ©) = 3))
82 fveq1 6890 . . . . . . . . . . . . . . . . 17 (𝑝 = βŸ¨β€œπ΄π΅πΆπ΄β€βŸ© β†’ (π‘β€˜0) = (βŸ¨β€œπ΄π΅πΆπ΄β€βŸ©β€˜0))
8382eqeq1d 2727 . . . . . . . . . . . . . . . 16 (𝑝 = βŸ¨β€œπ΄π΅πΆπ΄β€βŸ© β†’ ((π‘β€˜0) = 𝐴 ↔ (βŸ¨β€œπ΄π΅πΆπ΄β€βŸ©β€˜0) = 𝐴))
8483adantl 480 . . . . . . . . . . . . . . 15 ((𝑓 = βŸ¨β€œπ‘–π‘—π‘˜β€βŸ© ∧ 𝑝 = βŸ¨β€œπ΄π΅πΆπ΄β€βŸ©) β†’ ((π‘β€˜0) = 𝐴 ↔ (βŸ¨β€œπ΄π΅πΆπ΄β€βŸ©β€˜0) = 𝐴))
8579, 81, 843anbi123d 1432 . . . . . . . . . . . . . 14 ((𝑓 = βŸ¨β€œπ‘–π‘—π‘˜β€βŸ© ∧ 𝑝 = βŸ¨β€œπ΄π΅πΆπ΄β€βŸ©) β†’ ((𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴) ↔ (βŸ¨β€œπ‘–π‘—π‘˜β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΅πΆπ΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘–π‘—π‘˜β€βŸ©) = 3 ∧ (βŸ¨β€œπ΄π΅πΆπ΄β€βŸ©β€˜0) = 𝐴)))
8676, 78, 85spc2ev 3587 . . . . . . . . . . . . 13 ((βŸ¨β€œπ‘–π‘—π‘˜β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΅πΆπ΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘–π‘—π‘˜β€βŸ©) = 3 ∧ (βŸ¨β€œπ΄π΅πΆπ΄β€βŸ©β€˜0) = 𝐴) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴))
8774, 86syl 17 . . . . . . . . . . . 12 (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) ∧ ((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴))
8887expcom 412 . . . . . . . . . . 11 (((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) ∧ (π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) ∧ (𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴)))
89883exp 1116 . . . . . . . . . 10 ((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)) β†’ ((π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) β†’ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴)))))
9089rexlimiva 3137 . . . . . . . . 9 (βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—) β†’ ((π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) β†’ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴)))))
9190com12 32 . . . . . . . 8 ((π‘˜ ∈ dom (iEdgβ€˜πΊ) ∧ {𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) β†’ (βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—) β†’ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴)))))
9291rexlimiva 3137 . . . . . . 7 (βˆƒπ‘˜ ∈ dom (iEdgβ€˜πΊ){𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜) β†’ (βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—) β†’ ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴)))))
9392com13 88 . . . . . 6 ((𝑖 ∈ dom (iEdgβ€˜πΊ) ∧ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ (βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—) β†’ (βˆƒπ‘˜ ∈ dom (iEdgβ€˜πΊ){𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜) β†’ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴)))))
9493rexlimiva 3137 . . . . 5 (βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–) β†’ (βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—) β†’ (βˆƒπ‘˜ ∈ dom (iEdgβ€˜πΊ){𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜) β†’ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴)))))
95943imp 1108 . . . 4 ((βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–) ∧ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ βˆƒπ‘˜ ∈ dom (iEdgβ€˜πΊ){𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) β†’ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴)))
9695com12 32 . . 3 ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ ((βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–) ∧ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ βˆƒπ‘˜ ∈ dom (iEdgβ€˜πΊ){𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴)))
9713, 96sylbid 239 . 2 ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢))) β†’ (({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸 ∧ {𝐢, 𝐴} ∈ 𝐸) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴)))
98973impia 1114 1 ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝐴 β‰  𝐡 ∧ 𝐴 β‰  𝐢 ∧ 𝐡 β‰  𝐢)) ∧ ({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸 ∧ {𝐢, 𝐴} ∈ 𝐸)) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3 ∧ (π‘β€˜0) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098   β‰  wne 2930  βˆƒwrex 3060  Vcvv 3463   βŠ† wss 3940  {cpr 4626   class class class wbr 5143  dom cdm 5672  β€˜cfv 6542  0cc0 11136  3c3 12296  β™―chash 14319  Word cword 14494  βŸ¨β€œcs3 14823  βŸ¨β€œcs4 14824  Vtxcvtx 28851  iEdgciedg 28852  Edgcedg 28902  UHGraphcuhgr 28911  Cyclesccycls 29641
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-er 8721  df-map 8843  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-n0 12501  df-z 12587  df-uz 12851  df-fz 13515  df-fzo 13658  df-hash 14320  df-word 14495  df-concat 14551  df-s1 14576  df-s2 14829  df-s3 14830  df-s4 14831  df-edg 28903  df-uhgr 28913  df-wlks 29455  df-trls 29548  df-pths 29572  df-cycls 29643
This theorem is referenced by:  umgr3cyclex  30035
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