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| Mirrors > Home > MPE Home > Th. List > eupthf1o | Structured version Visualization version GIF version | ||
| Description: The 𝐹 function in an Eulerian path is a bijection from a half-open range of nonnegative integers to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) | 
| Ref | Expression | 
|---|---|
| eupths.i | ⊢ 𝐼 = (iEdg‘𝐺) | 
| Ref | Expression | 
|---|---|
| eupthf1o | ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eupths.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
| 2 | 1 | eupthi 30223 | . 2 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)) | 
| 3 | 2 | simprd 495 | 1 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 class class class wbr 5142 dom cdm 5684 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 0cc0 11156 ..^cfzo 13695 ♯chash 14370 iEdgciedg 29015 Walkscwlks 29615 EulerPathsceupth 30217 | 
| 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-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-wlks 29618 df-trls 29711 df-eupth 30218 | 
| This theorem is referenced by: eupthfi 30225 eupthvdres 30255 eucrctshift 30263 | 
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