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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 29147 | . 2 β’ (EulerPathsβπΊ) = {β¨π, πβ© β£ (π(TrailsβπΊ)π β§ π:(0..^(β―βπ))βontoβdom πΌ)} |
3 | simpl 484 | . . 3 β’ ((π = πΉ β§ π = π) β π = πΉ) | |
4 | fveq2 6843 | . . . . 5 β’ (π = πΉ β (β―βπ) = (β―βπΉ)) | |
5 | 4 | oveq2d 7374 | . . . 4 β’ (π = πΉ β (0..^(β―βπ)) = (0..^(β―βπΉ))) |
6 | 5 | adantr 482 | . . 3 β’ ((π = πΉ β§ π = π) β (0..^(β―βπ)) = (0..^(β―βπΉ))) |
7 | eqidd 2738 | . . 3 β’ ((π = πΉ β§ π = π) β dom πΌ = dom πΌ) | |
8 | 3, 6, 7 | foeq123d 6778 | . 2 β’ ((π = πΉ β§ π = π) β (π:(0..^(β―βπ))βontoβdom πΌ β πΉ:(0..^(β―βπΉ))βontoβdom πΌ)) |
9 | reltrls 28645 | . 2 β’ Rel (TrailsβπΊ) | |
10 | 2, 8, 9 | brfvopabrbr 6946 | 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 397 = wceq 1542 class class class wbr 5106 dom cdm 5634 βontoβwfo 6495 βcfv 6497 (class class class)co 7358 0cc0 11052 ..^cfzo 13568 β―chash 14231 iEdgciedg 27951 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-fo 6503 df-fv 6505 df-ov 7361 df-trls 28643 df-eupth 29145 |
This theorem is referenced by: iseupthf1o 29149 eupthistrl 29158 eucrctshift 29190 |
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