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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 30033 | . 2 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)) |
3 | 2 | simprd 494 | 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 5152 dom cdm 5682 –1-1-onto→wf1o 6552 ‘cfv 6553 (class class class)co 7426 0cc0 11146 ..^cfzo 13667 ♯chash 14329 iEdgciedg 28830 Walkscwlks 29430 EulerPathsceupth 30027 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-wlks 29433 df-trls 29526 df-eupth 30028 |
This theorem is referenced by: eupthfi 30035 eupthvdres 30065 eucrctshift 30073 |
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