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Mirrors > Home > MPE Home > Th. List > trlsfval | Structured version Visualization version GIF version |
Description: The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.) |
Ref | Expression |
---|---|
trlsfval | β’ (TrailsβπΊ) = {β¨π, πβ© β£ (π(WalksβπΊ)π β§ Fun β‘π)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 262 | . 2 β’ (π = πΊ β (Fun β‘π β Fun β‘π)) | |
2 | df-trls 29453 | . 2 β’ Trails = (π β V β¦ {β¨π, πβ© β£ (π(Walksβπ)π β§ Fun β‘π)}) | |
3 | 1, 2 | fvmptopab 7458 | 1 β’ (TrailsβπΊ) = {β¨π, πβ© β£ (π(WalksβπΊ)π β§ Fun β‘π)} |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 395 = wceq 1533 class class class wbr 5141 {copab 5203 β‘ccnv 5668 Fun wfun 6530 βcfv 6536 Walkscwlks 29357 Trailsctrls 29451 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-trls 29453 |
This theorem is referenced by: istrl 29457 upgrtrls 29462 |
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