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| Mirrors > Home > MPE Home > Th. List > iscycl | Structured version Visualization version GIF version | ||
| Description: Sufficient and necessary conditions for a pair of functions to be a cycle (in an undirected graph): A pair of function "is" (represents) a cycle iff it is a closed path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
| Ref | Expression |
|---|---|
| iscycl | ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cycls 29809 | . 2 ⊢ (Cycles‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))} | |
| 2 | fveq1 6905 | . . . 4 ⊢ (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0)) | |
| 3 | 2 | adantl 481 | . . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑝‘0) = (𝑃‘0)) |
| 4 | simpr 484 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃) | |
| 5 | fveq2 6906 | . . . . 5 ⊢ (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹)) | |
| 6 | 5 | adantr 480 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (♯‘𝑓) = (♯‘𝐹)) |
| 7 | 4, 6 | fveq12d 6913 | . . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑝‘(♯‘𝑓)) = (𝑃‘(♯‘𝐹))) |
| 8 | 3, 7 | eqeq12d 2753 | . 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) |
| 9 | relpths 29738 | . 2 ⊢ Rel (Paths‘𝐺) | |
| 10 | 1, 8, 9 | brfvopabrbr 7013 | 1 ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 class class class wbr 5143 ‘cfv 6561 0cc0 11155 ♯chash 14369 Pathscpths 29730 Cyclesccycls 29805 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-pths 29734 df-cycls 29807 |
| This theorem is referenced by: cyclprop 29813 cycliscrct 29819 cyclnumvtx 29820 cyclnspth 29821 pthisspthorcycl 29822 cyclispthon 29824 0cycl 30153 lp1cycl 30171 3cycld 30197 spthcycl 35134 subgrcycl 35140 2cycld 35143 |
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