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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 261 | . 2 β’ (π = πΊ β (Fun β‘π β Fun β‘π)) | |
2 | df-trls 29526 | . 2 β’ Trails = (π β V β¦ {β¨π, πβ© β£ (π(Walksβπ)π β§ Fun β‘π)}) | |
3 | 1, 2 | fvmptopab 7480 | 1 β’ (TrailsβπΊ) = {β¨π, πβ© β£ (π(WalksβπΊ)π β§ Fun β‘π)} |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 394 = wceq 1533 class class class wbr 5152 {copab 5214 β‘ccnv 5681 Fun wfun 6547 βcfv 6553 Walkscwlks 29430 Trailsctrls 29524 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-trls 29526 |
This theorem is referenced by: istrl 29530 upgrtrls 29535 |
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