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Mirrors > Home > MPE Home > Th. List > eupths | Structured version Visualization version GIF version |
Description: The Eulerian paths on the graph πΊ. (Contributed by AV, 18-Feb-2021.) (Revised by AV, 29-Oct-2021.) |
Ref | Expression |
---|---|
eupths.i | β’ πΌ = (iEdgβπΊ) |
Ref | Expression |
---|---|
eupths | β’ (EulerPathsβπΊ) = {β¨π, πβ© β£ (π(TrailsβπΊ)π β§ π:(0..^(β―βπ))βontoβdom πΌ)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6897 | . . . . 5 β’ (π = πΊ β (iEdgβπ) = (iEdgβπΊ)) | |
2 | eupths.i | . . . . 5 β’ πΌ = (iEdgβπΊ) | |
3 | 1, 2 | eqtr4di 2786 | . . . 4 β’ (π = πΊ β (iEdgβπ) = πΌ) |
4 | 3 | dmeqd 5908 | . . 3 β’ (π = πΊ β dom (iEdgβπ) = dom πΌ) |
5 | foeq3 6809 | . . 3 β’ (dom (iEdgβπ) = dom πΌ β (π:(0..^(β―βπ))βontoβdom (iEdgβπ) β π:(0..^(β―βπ))βontoβdom πΌ)) | |
6 | 4, 5 | syl 17 | . 2 β’ (π = πΊ β (π:(0..^(β―βπ))βontoβdom (iEdgβπ) β π:(0..^(β―βπ))βontoβdom πΌ)) |
7 | df-eupth 30021 | . 2 β’ EulerPaths = (π β V β¦ {β¨π, πβ© β£ (π(Trailsβπ)π β§ π:(0..^(β―βπ))βontoβdom (iEdgβπ))}) | |
8 | 6, 7 | fvmptopab 7474 | 1 β’ (EulerPathsβπΊ) = {β¨π, πβ© β£ (π(TrailsβπΊ)π β§ π:(0..^(β―βπ))βontoβdom πΌ)} |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 395 = wceq 1534 class class class wbr 5148 {copab 5210 dom cdm 5678 βontoβwfo 6546 βcfv 6548 (class class class)co 7420 0cc0 11139 ..^cfzo 13660 β―chash 14322 iEdgciedg 28823 Trailsctrls 29517 EulerPathsceupth 30020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fo 6554 df-fv 6556 df-eupth 30021 |
This theorem is referenced by: iseupth 30024 |
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