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| Mirrors > Home > MPE Home > Th. List > isclwlk | Structured version Visualization version GIF version | ||
| Description: A pair of functions represents a closed walk iff it represents a walk in which the first vertex is equal to the last vertex. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) | 
| Ref | Expression | 
|---|---|
| isclwlk | ⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | clwlks 29793 | . 2 ⊢ (ClWalks‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))} | |
| 2 | fveq1 6904 | . . . 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 6905 | . . . . 5 ⊢ (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹)) | |
| 6 | 5 | adantr 480 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (♯‘𝑓) = (♯‘𝐹)) | 
| 7 | 4, 6 | fveq12d 6912 | . . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑝‘(♯‘𝑓)) = (𝑃‘(♯‘𝐹))) | 
| 8 | 3, 7 | eqeq12d 2752 | . 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) | 
| 9 | relwlk 29645 | . 2 ⊢ Rel (Walks‘𝐺) | |
| 10 | 1, 8, 9 | brfvopabrbr 7012 | 1 ⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 class class class wbr 5142 ‘cfv 6560 0cc0 11156 ♯chash 14370 Walkscwlks 29615 ClWalkscclwlks 29791 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 df-wlks 29618 df-clwlks 29792 | 
| This theorem is referenced by: clwlkiswlk 29795 isclwlke 29798 isclwlkupgr 29799 clwlkcompbp 29803 clwlkl1loop 29804 crctisclwlk 29815 clwlkclwwlkflem 30024 clwlkclwwlkf 30028 0clwlk 30150 | 
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