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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 29960 | . 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 1533 class class class wbr 5141 dom cdm 5669 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7404 0cc0 11109 ..^cfzo 13630 ♯chash 14292 iEdgciedg 28760 Walkscwlks 29357 EulerPathsceupth 29954 |
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-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-ral 3056 df-rex 3065 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-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 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-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-wlks 29360 df-trls 29453 df-eupth 29955 |
This theorem is referenced by: eupthfi 29962 eupthvdres 29992 eucrctshift 30000 |
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