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Theorem iseupthf1o 29149
Description: The property "โŸจ๐น, ๐‘ƒโŸฉ is an Eulerian path on the graph ๐บ". An Eulerian path is defined as bijection ๐น from the edges to a set 0...(๐‘ โˆ’ 1) and a function ๐‘ƒ:(0...๐‘)โŸถ๐‘‰ into the vertices such that for each 0 โ‰ค ๐‘˜ < ๐‘, ๐น(๐‘˜) is an edge from ๐‘ƒ(๐‘˜) to ๐‘ƒ(๐‘˜ + 1). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.)
Hypothesis
Ref Expression
eupths.i ๐ผ = (iEdgโ€˜๐บ)
Assertion
Ref Expression
iseupthf1o (๐น(EulerPathsโ€˜๐บ)๐‘ƒ โ†” (๐น(Walksโ€˜๐บ)๐‘ƒ โˆง ๐น:(0..^(โ™ฏโ€˜๐น))โ€“1-1-ontoโ†’dom ๐ผ))

Proof of Theorem iseupthf1o
StepHypRef Expression
1 eupths.i . . 3 ๐ผ = (iEdgโ€˜๐บ)
21iseupth 29148 . 2 (๐น(EulerPathsโ€˜๐บ)๐‘ƒ โ†” (๐น(Trailsโ€˜๐บ)๐‘ƒ โˆง ๐น:(0..^(โ™ฏโ€˜๐น))โ€“ontoโ†’dom ๐ผ))
3 istrl 28647 . . . 4 (๐น(Trailsโ€˜๐บ)๐‘ƒ โ†” (๐น(Walksโ€˜๐บ)๐‘ƒ โˆง Fun โ—ก๐น))
43anbi1i 625 . . 3 ((๐น(Trailsโ€˜๐บ)๐‘ƒ โˆง ๐น:(0..^(โ™ฏโ€˜๐น))โ€“ontoโ†’dom ๐ผ) โ†” ((๐น(Walksโ€˜๐บ)๐‘ƒ โˆง Fun โ—ก๐น) โˆง ๐น:(0..^(โ™ฏโ€˜๐น))โ€“ontoโ†’dom ๐ผ))
5 anass 470 . . 3 (((๐น(Walksโ€˜๐บ)๐‘ƒ โˆง Fun โ—ก๐น) โˆง ๐น:(0..^(โ™ฏโ€˜๐น))โ€“ontoโ†’dom ๐ผ) โ†” (๐น(Walksโ€˜๐บ)๐‘ƒ โˆง (Fun โ—ก๐น โˆง ๐น:(0..^(โ™ฏโ€˜๐น))โ€“ontoโ†’dom ๐ผ)))
6 ancom 462 . . . 4 ((Fun โ—ก๐น โˆง ๐น:(0..^(โ™ฏโ€˜๐น))โ€“ontoโ†’dom ๐ผ) โ†” (๐น:(0..^(โ™ฏโ€˜๐น))โ€“ontoโ†’dom ๐ผ โˆง Fun โ—ก๐น))
76anbi2i 624 . . 3 ((๐น(Walksโ€˜๐บ)๐‘ƒ โˆง (Fun โ—ก๐น โˆง ๐น:(0..^(โ™ฏโ€˜๐น))โ€“ontoโ†’dom ๐ผ)) โ†” (๐น(Walksโ€˜๐บ)๐‘ƒ โˆง (๐น:(0..^(โ™ฏโ€˜๐น))โ€“ontoโ†’dom ๐ผ โˆง Fun โ—ก๐น)))
84, 5, 73bitri 297 . 2 ((๐น(Trailsโ€˜๐บ)๐‘ƒ โˆง ๐น:(0..^(โ™ฏโ€˜๐น))โ€“ontoโ†’dom ๐ผ) โ†” (๐น(Walksโ€˜๐บ)๐‘ƒ โˆง (๐น:(0..^(โ™ฏโ€˜๐น))โ€“ontoโ†’dom ๐ผ โˆง Fun โ—ก๐น)))
9 dff1o3 6791 . . . 4 (๐น:(0..^(โ™ฏโ€˜๐น))โ€“1-1-ontoโ†’dom ๐ผ โ†” (๐น:(0..^(โ™ฏโ€˜๐น))โ€“ontoโ†’dom ๐ผ โˆง Fun โ—ก๐น))
109bicomi 223 . . 3 ((๐น:(0..^(โ™ฏโ€˜๐น))โ€“ontoโ†’dom ๐ผ โˆง Fun โ—ก๐น) โ†” ๐น:(0..^(โ™ฏโ€˜๐น))โ€“1-1-ontoโ†’dom ๐ผ)
1110anbi2i 624 . 2 ((๐น(Walksโ€˜๐บ)๐‘ƒ โˆง (๐น:(0..^(โ™ฏโ€˜๐น))โ€“ontoโ†’dom ๐ผ โˆง Fun โ—ก๐น)) โ†” (๐น(Walksโ€˜๐บ)๐‘ƒ โˆง ๐น:(0..^(โ™ฏโ€˜๐น))โ€“1-1-ontoโ†’dom ๐ผ))
122, 8, 113bitri 297 1 (๐น(EulerPathsโ€˜๐บ)๐‘ƒ โ†” (๐น(Walksโ€˜๐บ)๐‘ƒ โˆง ๐น:(0..^(โ™ฏโ€˜๐น))โ€“1-1-ontoโ†’dom ๐ผ))
Colors of variables: wff setvar class
Syntax hints:   โ†” wb 205   โˆง wa 397   = wceq 1542   class class class wbr 5106  โ—กccnv 5633  dom cdm 5634  Fun wfun 6491  โ€“ontoโ†’wfo 6495  โ€“1-1-ontoโ†’wf1o 6496  โ€˜cfv 6497  (class class class)co 7358  0cc0 11052  ..^cfzo 13568  โ™ฏchash 14231  iEdgciedg 27951  Walkscwlks 28547  Trailsctrls 28641  EulerPathsceupth 29144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-wlks 28550  df-trls 28643  df-eupth 29145
This theorem is referenced by:  eupthi  29150  upgriseupth  29154  eupth0  29161  eupthres  29162  eupthp1  29163
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