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Mirrors > Home > MPE Home > Th. List > iseupth | Structured version Visualization version GIF version |
Description: The property "β¨πΉ, πβ© is an Eulerian path on the graph πΊ". An Eulerian path is defined as bijection πΉ from the edges to a set 0...(π β 1) and a function π:(0...π)βΆπ into the vertices such that for each 0 β€ π < π, πΉ(π) is an edge from π(π) to π(π + 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.) |
Ref | Expression |
---|---|
eupths.i | β’ πΌ = (iEdgβπΊ) |
Ref | Expression |
---|---|
iseupth | β’ (πΉ(EulerPathsβπΊ)π β (πΉ(TrailsβπΊ)π β§ πΉ:(0..^(β―βπΉ))βontoβdom πΌ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eupths.i | . . 3 β’ πΌ = (iEdgβπΊ) | |
2 | 1 | eupths 29958 | . 2 β’ (EulerPathsβπΊ) = {β¨π, πβ© β£ (π(TrailsβπΊ)π β§ π:(0..^(β―βπ))βontoβdom πΌ)} |
3 | simpl 482 | . . 3 β’ ((π = πΉ β§ π = π) β π = πΉ) | |
4 | fveq2 6884 | . . . . 5 β’ (π = πΉ β (β―βπ) = (β―βπΉ)) | |
5 | 4 | oveq2d 7420 | . . . 4 β’ (π = πΉ β (0..^(β―βπ)) = (0..^(β―βπΉ))) |
6 | 5 | adantr 480 | . . 3 β’ ((π = πΉ β§ π = π) β (0..^(β―βπ)) = (0..^(β―βπΉ))) |
7 | eqidd 2727 | . . 3 β’ ((π = πΉ β§ π = π) β dom πΌ = dom πΌ) | |
8 | 3, 6, 7 | foeq123d 6819 | . 2 β’ ((π = πΉ β§ π = π) β (π:(0..^(β―βπ))βontoβdom πΌ β πΉ:(0..^(β―βπΉ))βontoβdom πΌ)) |
9 | reltrls 29456 | . 2 β’ Rel (TrailsβπΊ) | |
10 | 2, 8, 9 | brfvopabrbr 6988 | 1 β’ (πΉ(EulerPathsβπΊ)π β (πΉ(TrailsβπΊ)π β§ πΉ:(0..^(β―βπΉ))βontoβdom πΌ)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 395 = wceq 1533 class class class wbr 5141 dom cdm 5669 βontoβwfo 6534 βcfv 6536 (class class class)co 7404 0cc0 11109 ..^cfzo 13630 β―chash 14293 iEdgciedg 28761 Trailsctrls 29452 EulerPathsceupth 29955 |
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-fo 6542 df-fv 6544 df-ov 7407 df-trls 29454 df-eupth 29956 |
This theorem is referenced by: iseupthf1o 29960 eupthistrl 29969 eucrctshift 30001 |
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