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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 6843 | . . . . 5 β’ (π = πΊ β (iEdgβπ) = (iEdgβπΊ)) | |
2 | eupths.i | . . . . 5 β’ πΌ = (iEdgβπΊ) | |
3 | 1, 2 | eqtr4di 2795 | . . . 4 β’ (π = πΊ β (iEdgβπ) = πΌ) |
4 | 3 | dmeqd 5862 | . . 3 β’ (π = πΊ β dom (iEdgβπ) = dom πΌ) |
5 | foeq3 6755 | . . 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 29145 | . 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 7412 | 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 397 = wceq 1542 class class class wbr 5106 {copab 5168 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-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-iota 6449 df-fun 6499 df-fo 6503 df-fv 6505 df-eupth 29145 |
This theorem is referenced by: iseupth 29148 |
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