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Mirrors > Home > MPE Home > Th. List > iseupthf1o | Structured version Visualization version GIF version |
Description: The property "โจ๐น, ๐โฉ is an Eulerian path on the graph ๐บ". An Eulerian path is defined as bijection ๐น from the edges to a set 0...(๐ โ 1) and a function ๐:(0...๐)โถ๐ into the vertices such that for each 0 โค ๐ < ๐, ๐น(๐) is an edge from ๐(๐) to ๐(๐ + 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 | โข ๐ผ = (iEdgโ๐บ) |
Ref | Expression |
---|---|
iseupthf1o | โข (๐น(EulerPathsโ๐บ)๐ โ (๐น(Walksโ๐บ)๐ โง ๐น:(0..^(โฏโ๐น))โ1-1-ontoโdom ๐ผ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eupths.i | . . 3 โข ๐ผ = (iEdgโ๐บ) | |
2 | 1 | iseupth 29148 | . 2 โข (๐น(EulerPathsโ๐บ)๐ โ (๐น(Trailsโ๐บ)๐ โง ๐น:(0..^(โฏโ๐น))โontoโdom ๐ผ)) |
3 | istrl 28647 | . . . 4 โข (๐น(Trailsโ๐บ)๐ โ (๐น(Walksโ๐บ)๐ โง Fun โก๐น)) | |
4 | 3 | anbi1i 625 | . . 3 โข ((๐น(Trailsโ๐บ)๐ โง ๐น:(0..^(โฏโ๐น))โontoโdom ๐ผ) โ ((๐น(Walksโ๐บ)๐ โง Fun โก๐น) โง ๐น:(0..^(โฏโ๐น))โontoโdom ๐ผ)) |
5 | anass 470 | . . 3 โข (((๐น(Walksโ๐บ)๐ โง Fun โก๐น) โง ๐น:(0..^(โฏโ๐น))โontoโdom ๐ผ) โ (๐น(Walksโ๐บ)๐ โง (Fun โก๐น โง ๐น:(0..^(โฏโ๐น))โontoโdom ๐ผ))) | |
6 | ancom 462 | . . . 4 โข ((Fun โก๐น โง ๐น:(0..^(โฏโ๐น))โontoโdom ๐ผ) โ (๐น:(0..^(โฏโ๐น))โontoโdom ๐ผ โง Fun โก๐น)) | |
7 | 6 | anbi2i 624 | . . 3 โข ((๐น(Walksโ๐บ)๐ โง (Fun โก๐น โง ๐น:(0..^(โฏโ๐น))โontoโdom ๐ผ)) โ (๐น(Walksโ๐บ)๐ โง (๐น:(0..^(โฏโ๐น))โontoโdom ๐ผ โง Fun โก๐น))) |
8 | 4, 5, 7 | 3bitri 297 | . 2 โข ((๐น(Trailsโ๐บ)๐ โง ๐น:(0..^(โฏโ๐น))โontoโdom ๐ผ) โ (๐น(Walksโ๐บ)๐ โง (๐น:(0..^(โฏโ๐น))โontoโdom ๐ผ โง Fun โก๐น))) |
9 | dff1o3 6791 | . . . 4 โข (๐น:(0..^(โฏโ๐น))โ1-1-ontoโdom ๐ผ โ (๐น:(0..^(โฏโ๐น))โontoโdom ๐ผ โง Fun โก๐น)) | |
10 | 9 | bicomi 223 | . . 3 โข ((๐น:(0..^(โฏโ๐น))โontoโdom ๐ผ โง Fun โก๐น) โ ๐น:(0..^(โฏโ๐น))โ1-1-ontoโdom ๐ผ) |
11 | 10 | anbi2i 624 | . 2 โข ((๐น(Walksโ๐บ)๐ โง (๐น:(0..^(โฏโ๐น))โontoโdom ๐ผ โง Fun โก๐น)) โ (๐น(Walksโ๐บ)๐ โง ๐น:(0..^(โฏโ๐น))โ1-1-ontoโdom ๐ผ)) |
12 | 2, 8, 11 | 3bitri 297 | 1 โข (๐น(EulerPathsโ๐บ)๐ โ (๐น(Walksโ๐บ)๐ โง ๐น:(0..^(โฏโ๐น))โ1-1-ontoโdom ๐ผ)) |
Colors of variables: wff setvar class |
Syntax hints: โ wb 205 โง wa 397 = wceq 1542 class class class wbr 5106 โกccnv 5633 dom cdm 5634 Fun wfun 6491 โontoโwfo 6495 โ1-1-ontoโwf1o 6496 โcfv 6497 (class class class)co 7358 0cc0 11052 ..^cfzo 13568 โฏchash 14231 iEdgciedg 27951 Walkscwlks 28547 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-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-wlks 28550 df-trls 28643 df-eupth 29145 |
This theorem is referenced by: eupthi 29150 upgriseupth 29154 eupth0 29161 eupthres 29162 eupthp1 29163 |
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