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Theorem eupthf1o 29095
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.)
Hypothesis
Ref Expression
eupths.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
eupthf1o (𝐹(EulerPaths‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)

Proof of Theorem eupthf1o
StepHypRef Expression
1 eupths.i . . 3 𝐼 = (iEdg‘𝐺)
21eupthi 29094 . 2 (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼))
32simprd 496 1 (𝐹(EulerPaths‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541   class class class wbr 5105  dom cdm 5633  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7356  0cc0 11050  ..^cfzo 13566  chash 14229  iEdgciedg 27895  Walkscwlks 28491  EulerPathsceupth 29088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7359  df-wlks 28494  df-trls 28587  df-eupth 29089
This theorem is referenced by:  eupthfi  29096  eupthvdres  29126  eucrctshift  29134
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