Users' Mathboxes Mathbox for BTernaryTau < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  loop1cycl Structured version   Visualization version   GIF version

Theorem loop1cycl 34656
Description: A hypergraph has a cycle of length one if and only if it has a loop. (Contributed by BTernaryTau, 13-Oct-2023.)
Assertion
Ref Expression
loop1cycl (𝐺 ∈ UHGraph β†’ (βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴) ↔ {𝐴} ∈ (Edgβ€˜πΊ)))
Distinct variable groups:   𝐺,𝑝   𝐴,𝑓,𝑝   𝑓,𝐺

Proof of Theorem loop1cycl
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 cyclprop 29559 . . . . . . . . . . . . 13 (𝑓(Cyclesβ€˜πΊ)𝑝 β†’ (𝑓(Pathsβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“))))
2 fveq2 6885 . . . . . . . . . . . . . . . 16 ((β™―β€˜π‘“) = 1 β†’ (π‘β€˜(β™―β€˜π‘“)) = (π‘β€˜1))
32eqeq2d 2737 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘“) = 1 β†’ ((π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“)) ↔ (π‘β€˜0) = (π‘β€˜1)))
43anbi2d 628 . . . . . . . . . . . . . 14 ((β™―β€˜π‘“) = 1 β†’ ((𝑓(Pathsβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“))) ↔ (𝑓(Pathsβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜1))))
54biimpd 228 . . . . . . . . . . . . 13 ((β™―β€˜π‘“) = 1 β†’ ((𝑓(Pathsβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“))) β†’ (𝑓(Pathsβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜1))))
61, 5mpan9 506 . . . . . . . . . . . 12 ((𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1) β†’ (𝑓(Pathsβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜1)))
7 pthiswlk 29493 . . . . . . . . . . . . 13 (𝑓(Pathsβ€˜πΊ)𝑝 β†’ 𝑓(Walksβ€˜πΊ)𝑝)
87anim1i 614 . . . . . . . . . . . 12 ((𝑓(Pathsβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜1)) β†’ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜1)))
96, 8syl 17 . . . . . . . . . . 11 ((𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1) β†’ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜1)))
109anim1i 614 . . . . . . . . . 10 (((𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1) ∧ (β™―β€˜π‘“) = 1) β†’ ((𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜1)) ∧ (β™―β€˜π‘“) = 1))
1110anabss3 672 . . . . . . . . 9 ((𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1) β†’ ((𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜1)) ∧ (β™―β€˜π‘“) = 1))
12 df-3an 1086 . . . . . . . . 9 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜1) ∧ (β™―β€˜π‘“) = 1) ↔ ((𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜1)) ∧ (β™―β€˜π‘“) = 1))
1311, 12sylibr 233 . . . . . . . 8 ((𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1) β†’ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜1) ∧ (β™―β€˜π‘“) = 1))
14 3ancomb 1096 . . . . . . . 8 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜1) ∧ (β™―β€˜π‘“) = 1) ↔ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = (π‘β€˜1)))
1513, 14sylib 217 . . . . . . 7 ((𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1) β†’ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = (π‘β€˜1)))
16 wlkl1loop 29404 . . . . . . . . . 10 (((Fun (iEdgβ€˜πΊ) ∧ 𝑓(Walksβ€˜πΊ)𝑝) ∧ ((β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = (π‘β€˜1))) β†’ {(π‘β€˜0)} ∈ (Edgβ€˜πΊ))
1716expl 457 . . . . . . . . 9 (Fun (iEdgβ€˜πΊ) β†’ ((𝑓(Walksβ€˜πΊ)𝑝 ∧ ((β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = (π‘β€˜1))) β†’ {(π‘β€˜0)} ∈ (Edgβ€˜πΊ)))
18 eqid 2726 . . . . . . . . . 10 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
1918uhgrfun 28834 . . . . . . . . 9 (𝐺 ∈ UHGraph β†’ Fun (iEdgβ€˜πΊ))
2017, 19syl11 33 . . . . . . . 8 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ ((β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = (π‘β€˜1))) β†’ (𝐺 ∈ UHGraph β†’ {(π‘β€˜0)} ∈ (Edgβ€˜πΊ)))
21203impb 1112 . . . . . . 7 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = (π‘β€˜1)) β†’ (𝐺 ∈ UHGraph β†’ {(π‘β€˜0)} ∈ (Edgβ€˜πΊ)))
2215, 21syl 17 . . . . . 6 ((𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1) β†’ (𝐺 ∈ UHGraph β†’ {(π‘β€˜0)} ∈ (Edgβ€˜πΊ)))
23223adant3 1129 . . . . 5 ((𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴) β†’ (𝐺 ∈ UHGraph β†’ {(π‘β€˜0)} ∈ (Edgβ€˜πΊ)))
24 sneq 4633 . . . . . . 7 ((π‘β€˜0) = 𝐴 β†’ {(π‘β€˜0)} = {𝐴})
2524eleq1d 2812 . . . . . 6 ((π‘β€˜0) = 𝐴 β†’ ({(π‘β€˜0)} ∈ (Edgβ€˜πΊ) ↔ {𝐴} ∈ (Edgβ€˜πΊ)))
26253ad2ant3 1132 . . . . 5 ((𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴) β†’ ({(π‘β€˜0)} ∈ (Edgβ€˜πΊ) ↔ {𝐴} ∈ (Edgβ€˜πΊ)))
2723, 26sylibd 238 . . . 4 ((𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴) β†’ (𝐺 ∈ UHGraph β†’ {𝐴} ∈ (Edgβ€˜πΊ)))
2827exlimivv 1927 . . 3 (βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴) β†’ (𝐺 ∈ UHGraph β†’ {𝐴} ∈ (Edgβ€˜πΊ)))
2928com12 32 . 2 (𝐺 ∈ UHGraph β†’ (βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴) β†’ {𝐴} ∈ (Edgβ€˜πΊ)))
30 edgval 28817 . . . . . . . . . . . . . 14 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
3130eleq2i 2819 . . . . . . . . . . . . 13 ({𝐴} ∈ (Edgβ€˜πΊ) ↔ {𝐴} ∈ ran (iEdgβ€˜πΊ))
32 elrnrexdm 7084 . . . . . . . . . . . . . 14 (Fun (iEdgβ€˜πΊ) β†’ ({𝐴} ∈ ran (iEdgβ€˜πΊ) β†’ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐴} = ((iEdgβ€˜πΊ)β€˜π‘—)))
33 eqcom 2733 . . . . . . . . . . . . . . 15 ({𝐴} = ((iEdgβ€˜πΊ)β€˜π‘—) ↔ ((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴})
3433rexbii 3088 . . . . . . . . . . . . . 14 (βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐴} = ((iEdgβ€˜πΊ)β€˜π‘—) ↔ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ)((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴})
3532, 34imbitrdi 250 . . . . . . . . . . . . 13 (Fun (iEdgβ€˜πΊ) β†’ ({𝐴} ∈ ran (iEdgβ€˜πΊ) β†’ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ)((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴}))
3631, 35biimtrid 241 . . . . . . . . . . . 12 (Fun (iEdgβ€˜πΊ) β†’ ({𝐴} ∈ (Edgβ€˜πΊ) β†’ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ)((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴}))
3719, 36syl 17 . . . . . . . . . . 11 (𝐺 ∈ UHGraph β†’ ({𝐴} ∈ (Edgβ€˜πΊ) β†’ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ)((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴}))
38 df-rex 3065 . . . . . . . . . . 11 (βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ)((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴} ↔ βˆƒπ‘—(𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ ((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴}))
3937, 38imbitrdi 250 . . . . . . . . . 10 (𝐺 ∈ UHGraph β†’ ({𝐴} ∈ (Edgβ€˜πΊ) β†’ βˆƒπ‘—(𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ ((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴})))
4018lp1cycl 29914 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ 𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ ((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴}) β†’ βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ©)
41403expib 1119 . . . . . . . . . . 11 (𝐺 ∈ UHGraph β†’ ((𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ ((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴}) β†’ βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ©))
4241eximdv 1912 . . . . . . . . . 10 (𝐺 ∈ UHGraph β†’ (βˆƒπ‘—(𝑗 ∈ dom (iEdgβ€˜πΊ) ∧ ((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴}) β†’ βˆƒπ‘—βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ©))
4339, 42syld 47 . . . . . . . . 9 (𝐺 ∈ UHGraph β†’ ({𝐴} ∈ (Edgβ€˜πΊ) β†’ βˆƒπ‘—βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ©))
44 s1len 14562 . . . . . . . . . . 11 (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1
4544ax-gen 1789 . . . . . . . . . 10 βˆ€π‘—(β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1
46 19.29r 1869 . . . . . . . . . 10 ((βˆƒπ‘—βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ βˆ€π‘—(β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1) β†’ βˆƒπ‘—(βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1))
4745, 46mpan2 688 . . . . . . . . 9 (βˆƒπ‘—βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© β†’ βˆƒπ‘—(βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1))
4843, 47syl6 35 . . . . . . . 8 (𝐺 ∈ UHGraph β†’ ({𝐴} ∈ (Edgβ€˜πΊ) β†’ βˆƒπ‘—(βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1)))
4948imp 406 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edgβ€˜πΊ)) β†’ βˆƒπ‘—(βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1))
50 uhgredgn0 28896 . . . . . . . . . . 11 ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edgβ€˜πΊ)) β†’ {𝐴} ∈ (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}))
51 eldifsni 4788 . . . . . . . . . . 11 ({𝐴} ∈ (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}) β†’ {𝐴} β‰  βˆ…)
5250, 51syl 17 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edgβ€˜πΊ)) β†’ {𝐴} β‰  βˆ…)
53 snnzb 4717 . . . . . . . . . 10 (𝐴 ∈ V ↔ {𝐴} β‰  βˆ…)
5452, 53sylibr 233 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edgβ€˜πΊ)) β†’ 𝐴 ∈ V)
55 s2fv0 14844 . . . . . . . . 9 (𝐴 ∈ V β†’ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴)
5654, 55syl 17 . . . . . . . 8 ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edgβ€˜πΊ)) β†’ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴)
5756alrimiv 1922 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edgβ€˜πΊ)) β†’ βˆ€π‘—(βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴)
58 19.29r 1869 . . . . . . 7 ((βˆƒπ‘—(βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1) ∧ βˆ€π‘—(βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴) β†’ βˆƒπ‘—((βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1) ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴))
5949, 57, 58syl2anc 583 . . . . . 6 ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edgβ€˜πΊ)) β†’ βˆƒπ‘—((βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1) ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴))
60 df-3an 1086 . . . . . . 7 ((βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴) ↔ ((βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1) ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴))
6160exbii 1842 . . . . . 6 (βˆƒπ‘—(βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴) ↔ βˆƒπ‘—((βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1) ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴))
6259, 61sylibr 233 . . . . 5 ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edgβ€˜πΊ)) β†’ βˆƒπ‘—(βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴))
63 s1cli 14561 . . . . . . . 8 βŸ¨β€œπ‘—β€βŸ© ∈ Word V
64 breq1 5144 . . . . . . . . . 10 (𝑓 = βŸ¨β€œπ‘—β€βŸ© β†’ (𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ↔ βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ©))
65 fveqeq2 6894 . . . . . . . . . 10 (𝑓 = βŸ¨β€œπ‘—β€βŸ© β†’ ((β™―β€˜π‘“) = 1 ↔ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1))
6664, 653anbi12d 1433 . . . . . . . . 9 (𝑓 = βŸ¨β€œπ‘—β€βŸ© β†’ ((𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜π‘“) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴) ↔ (βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴)))
6766rspcev 3606 . . . . . . . 8 ((βŸ¨β€œπ‘—β€βŸ© ∈ Word V ∧ (βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴)) β†’ βˆƒπ‘“ ∈ Word V(𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜π‘“) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴))
6863, 67mpan 687 . . . . . . 7 ((βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴) β†’ βˆƒπ‘“ ∈ Word V(𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜π‘“) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴))
69 rexex 3070 . . . . . . 7 (βˆƒπ‘“ ∈ Word V(𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜π‘“) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴) β†’ βˆƒπ‘“(𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜π‘“) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴))
7068, 69syl 17 . . . . . 6 ((βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴) β†’ βˆƒπ‘“(𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜π‘“) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴))
7170exlimiv 1925 . . . . 5 (βˆƒπ‘—(βŸ¨β€œπ‘—β€βŸ©(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—β€βŸ©) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴) β†’ βˆƒπ‘“(𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜π‘“) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴))
7262, 71syl 17 . . . 4 ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edgβ€˜πΊ)) β†’ βˆƒπ‘“(𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜π‘“) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴))
73 s2cli 14837 . . . . . . 7 βŸ¨β€œπ΄π΄β€βŸ© ∈ Word V
74 breq2 5145 . . . . . . . . 9 (𝑝 = βŸ¨β€œπ΄π΄β€βŸ© β†’ (𝑓(Cyclesβ€˜πΊ)𝑝 ↔ 𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ©))
75 fveq1 6884 . . . . . . . . . 10 (𝑝 = βŸ¨β€œπ΄π΄β€βŸ© β†’ (π‘β€˜0) = (βŸ¨β€œπ΄π΄β€βŸ©β€˜0))
7675eqeq1d 2728 . . . . . . . . 9 (𝑝 = βŸ¨β€œπ΄π΄β€βŸ© β†’ ((π‘β€˜0) = 𝐴 ↔ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴))
7774, 763anbi13d 1434 . . . . . . . 8 (𝑝 = βŸ¨β€œπ΄π΄β€βŸ© β†’ ((𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴) ↔ (𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜π‘“) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴)))
7877rspcev 3606 . . . . . . 7 ((βŸ¨β€œπ΄π΄β€βŸ© ∈ Word V ∧ (𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜π‘“) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴)) β†’ βˆƒπ‘ ∈ Word V(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴))
7973, 78mpan 687 . . . . . 6 ((𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜π‘“) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴) β†’ βˆƒπ‘ ∈ Word V(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴))
80 rexex 3070 . . . . . 6 (βˆƒπ‘ ∈ Word V(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴) β†’ βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴))
8179, 80syl 17 . . . . 5 ((𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜π‘“) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴) β†’ βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴))
8281eximi 1829 . . . 4 (βˆƒπ‘“(𝑓(Cyclesβ€˜πΊ)βŸ¨β€œπ΄π΄β€βŸ© ∧ (β™―β€˜π‘“) = 1 ∧ (βŸ¨β€œπ΄π΄β€βŸ©β€˜0) = 𝐴) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴))
8372, 82syl 17 . . 3 ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edgβ€˜πΊ)) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴))
8483ex 412 . 2 (𝐺 ∈ UHGraph β†’ ({𝐴} ∈ (Edgβ€˜πΊ) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴)))
8529, 84impbid 211 1 (𝐺 ∈ UHGraph β†’ (βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴) ↔ {𝐴} ∈ (Edgβ€˜πΊ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084  βˆ€wal 1531   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098   β‰  wne 2934  βˆƒwrex 3064  Vcvv 3468   βˆ– cdif 3940  βˆ…c0 4317  π’« cpw 4597  {csn 4623   class class class wbr 5141  dom cdm 5669  ran crn 5670  Fun wfun 6531  β€˜cfv 6537  0cc0 11112  1c1 11113  β™―chash 14295  Word cword 14470  βŸ¨β€œcs1 14551  βŸ¨β€œcs2 14798  Vtxcvtx 28764  iEdgciedg 28765  Edgcedg 28815  UHGraphcuhgr 28824  Walkscwlks 29362  Pathscpths 29478  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-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-pm 8825  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-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-edg 28816  df-uhgr 28826  df-wlks 29365  df-wlkson 29366  df-trls 29458  df-trlson 29459  df-pths 29482  df-pthson 29484  df-cycls 29553
This theorem is referenced by:  acycgrislfgr  34671
  Copyright terms: Public domain W3C validator