![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > istrl | Structured version Visualization version GIF version |
Description: Conditions for a pair of classes/functions to be a trail (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 |
---|---|
istrl | β’ (πΉ(TrailsβπΊ)π β (πΉ(WalksβπΊ)π β§ Fun β‘πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlsfval 28952 | . 2 β’ (TrailsβπΊ) = {β¨π, πβ© β£ (π(WalksβπΊ)π β§ Fun β‘π)} | |
2 | cnveq 5874 | . . . 4 β’ (π = πΉ β β‘π = β‘πΉ) | |
3 | 2 | funeqd 6571 | . . 3 β’ (π = πΉ β (Fun β‘π β Fun β‘πΉ)) |
4 | 3 | adantr 482 | . 2 β’ ((π = πΉ β§ π = π) β (Fun β‘π β Fun β‘πΉ)) |
5 | relwlk 28883 | . 2 β’ Rel (WalksβπΊ) | |
6 | 1, 4, 5 | brfvopabrbr 6996 | 1 β’ (πΉ(TrailsβπΊ)π β (πΉ(WalksβπΊ)π β§ Fun β‘πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 = wceq 1542 class class class wbr 5149 β‘ccnv 5676 Fun wfun 6538 βcfv 6544 Walkscwlks 28853 Trailsctrls 28947 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fv 6552 df-wlks 28856 df-trls 28949 |
This theorem is referenced by: trliswlk 28954 trlf1 28955 trlres 28957 upgristrl 28959 2pthnloop 28988 upgrspthswlk 28995 uhgrwkspth 29012 usgr2wlkspth 29016 uspgrn2crct 29062 crctcshtrl 29077 2trld 29192 0trl 29375 1trld 29395 ntrl2v2e 29411 3trld 29425 iseupthf1o 29455 subgrtrl 34124 |
Copyright terms: Public domain | W3C validator |