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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 6882 | . . . . 5 β’ (π = πΊ β (iEdgβπ) = (iEdgβπΊ)) | |
2 | eupths.i | . . . . 5 β’ πΌ = (iEdgβπΊ) | |
3 | 1, 2 | eqtr4di 2782 | . . . 4 β’ (π = πΊ β (iEdgβπ) = πΌ) |
4 | 3 | dmeqd 5896 | . . 3 β’ (π = πΊ β dom (iEdgβπ) = dom πΌ) |
5 | foeq3 6794 | . . 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 29946 | . 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 7456 | 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 1533 class class class wbr 5139 {copab 5201 dom cdm 5667 βontoβwfo 6532 βcfv 6534 (class class class)co 7402 0cc0 11107 ..^cfzo 13628 β―chash 14291 iEdgciedg 28751 Trailsctrls 29442 EulerPathsceupth 29945 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6486 df-fun 6536 df-fo 6540 df-fv 6542 df-eupth 29946 |
This theorem is referenced by: iseupth 29949 |
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