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Theorem uhgr3cyclex 29944
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 2819 . . . . . 6 ({𝐴, 𝐡} ∈ 𝐸 ↔ {𝐴, 𝐡} ∈ (Edgβ€˜πΊ))
3 eqid 2726 . . . . . . 7 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
43uhgredgiedgb 28894 . . . . . 6 (𝐺 ∈ UHGraph β†’ ({𝐴, 𝐡} ∈ (Edgβ€˜πΊ) ↔ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))
52, 4bitrid 283 . . . . 5 (𝐺 ∈ UHGraph β†’ ({𝐴, 𝐡} ∈ 𝐸 ↔ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–)))
61eleq2i 2819 . . . . . 6 ({𝐡, 𝐢} ∈ 𝐸 ↔ {𝐡, 𝐢} ∈ (Edgβ€˜πΊ))
73uhgredgiedgb 28894 . . . . . 6 (𝐺 ∈ UHGraph β†’ ({𝐡, 𝐢} ∈ (Edgβ€˜πΊ) ↔ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)))
86, 7bitrid 283 . . . . 5 (𝐺 ∈ UHGraph β†’ ({𝐡, 𝐢} ∈ 𝐸 ↔ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—)))
91eleq2i 2819 . . . . . 6 ({𝐢, 𝐴} ∈ 𝐸 ↔ {𝐢, 𝐴} ∈ (Edgβ€˜πΊ))
103uhgredgiedgb 28894 . . . . . 6 (𝐺 ∈ UHGraph β†’ ({𝐢, 𝐴} ∈ (Edgβ€˜πΊ) ↔ βˆƒπ‘˜ ∈ dom (iEdgβ€˜πΊ){𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜)))
119, 10bitrid 283 . . . . 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 480 . . 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 2726 . . . . . . . . . . . . . 14 βŸ¨β€œπ΄π΅πΆπ΄β€βŸ© = βŸ¨β€œπ΄π΅πΆπ΄β€βŸ©
15 eqid 2726 . . . . . . . . . . . . . 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 pm3.22 459 . . . . . . . . . . . . . . . . . 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 511 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐢 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
2019adantr 480 . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . 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 2988 . . . . . . . . . . . . . . . . . . . 20 (𝐴 β‰  𝐡 ↔ 𝐡 β‰  𝐴)
2423biimpi 215 . . . . . . . . . . . . . . . . . . 19 (𝐴 β‰  𝐡 β†’ 𝐡 β‰  𝐴)
2524anim1ci 615 . . . . . . . . . . . . . . . . . 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 2988 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . 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 4035 . . . . . . . . . . . . . . . . . 18 ({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘–) β†’ {𝐴, 𝐡} βŠ† ((iEdgβ€˜πΊ)β€˜π‘–))
3433adantl 481 . . . . . . . . . . . . . . . . 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 4035 . . . . . . . . . . . . . . . . . 18 ({𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘—) β†’ {𝐡, 𝐢} βŠ† ((iEdgβ€˜πΊ)β€˜π‘—))
3736adantl 481 . . . . . . . . . . . . . . . . 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 4035 . . . . . . . . . . . . . . . . . 18 ({𝐢, 𝐴} = ((iEdgβ€˜πΊ)β€˜π‘˜) β†’ {𝐢, 𝐴} βŠ† ((iEdgβ€˜πΊ)β€˜π‘˜))
4039adantl 481 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 511 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (𝐢 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
4847, 29anim12i 612 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 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 pm3.22 459 . . . . . . . . . . . . . . . . 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 29943 . . . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . . . 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 612 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 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 29943 . . . . . . . . . . . . . . . . 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 2990 . . . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . . . 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 29943 . . . . . . . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . 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 2727 . . . . . . . . . . . . . 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 29941 . . . . . . . . . . . . 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 14838 . . . . . . . . . . . . . . 15 βŸ¨β€œπ‘–π‘—π‘˜β€βŸ© ∈ Word V
7675elexi 3488 . . . . . . . . . . . . . 14 βŸ¨β€œπ‘–π‘—π‘˜β€βŸ© ∈ V
77 s4cli 14839 . . . . . . . . . . . . . . 15 βŸ¨β€œπ΄π΅πΆπ΄β€βŸ© ∈ Word V
7877elexi 3488 . . . . . . . . . . . . . 14 βŸ¨β€œπ΄π΅πΆπ΄β€βŸ© ∈ V
79 breq12 5146 . . . . . . . . . . . . . . 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 6894 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 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 6884 . . . . . . . . . . . . . . . . 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 2728 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 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 3591 . . . . . . . . . . . . 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 413 . . . . . . . . . . 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 3141 . . . . . . . . 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 3141 . . . . . . 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 3141 . . . . 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 395   ∧ w3a 1084   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098   β‰  wne 2934  βˆƒwrex 3064  Vcvv 3468   βŠ† wss 3943  {cpr 4625   class class class wbr 5141  dom cdm 5669  β€˜cfv 6537  0cc0 11112  3c3 12272  β™―chash 14295  Word cword 14470  βŸ¨β€œcs3 14799  βŸ¨β€œcs4 14800  Vtxcvtx 28764  iEdgciedg 28765  Edgcedg 28815  UHGraphcuhgr 28824  Cyclesccycls 29551
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1060  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-hash 14296  df-word 14471  df-concat 14527  df-s1 14552  df-s2 14805  df-s3 14806  df-s4 14807  df-edg 28816  df-uhgr 28826  df-wlks 29365  df-trls 29458  df-pths 29482  df-cycls 29553
This theorem is referenced by:  umgr3cyclex  29945
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