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Mirrors > Home > MPE Home > Th. List > iscrct | Structured version Visualization version GIF version |
Description: Sufficient and necessary conditions for a pair of functions to be a circuit (in an undirected graph): A pair of function "is" (represents) a circuit iff it is a closed trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
iscrct | β’ (πΉ(CircuitsβπΊ)π β (πΉ(TrailsβπΊ)π β§ (πβ0) = (πβ(β―βπΉ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crcts 29312 | . 2 β’ (CircuitsβπΊ) = {β¨π, πβ© β£ (π(TrailsβπΊ)π β§ (πβ0) = (πβ(β―βπ)))} | |
2 | fveq1 6889 | . . . 4 β’ (π = π β (πβ0) = (πβ0)) | |
3 | 2 | adantl 480 | . . 3 β’ ((π = πΉ β§ π = π) β (πβ0) = (πβ0)) |
4 | simpr 483 | . . . 4 β’ ((π = πΉ β§ π = π) β π = π) | |
5 | fveq2 6890 | . . . . 5 β’ (π = πΉ β (β―βπ) = (β―βπΉ)) | |
6 | 5 | adantr 479 | . . . 4 β’ ((π = πΉ β§ π = π) β (β―βπ) = (β―βπΉ)) |
7 | 4, 6 | fveq12d 6897 | . . 3 β’ ((π = πΉ β§ π = π) β (πβ(β―βπ)) = (πβ(β―βπΉ))) |
8 | 3, 7 | eqeq12d 2746 | . 2 β’ ((π = πΉ β§ π = π) β ((πβ0) = (πβ(β―βπ)) β (πβ0) = (πβ(β―βπΉ)))) |
9 | reltrls 29218 | . 2 β’ Rel (TrailsβπΊ) | |
10 | 1, 8, 9 | brfvopabrbr 6994 | 1 β’ (πΉ(CircuitsβπΊ)π β (πΉ(TrailsβπΊ)π β§ (πβ0) = (πβ(β―βπΉ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 394 = wceq 1539 class class class wbr 5147 βcfv 6542 0cc0 11112 β―chash 14294 Trailsctrls 29214 Circuitsccrcts 29308 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fv 6550 df-trls 29216 df-crcts 29310 |
This theorem is referenced by: crctprop 29316 cycliscrct 29323 crctcsh 29345 0crct 29653 eupth2eucrct 29737 |
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