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Mirrors > Home > MPE Home > Th. List > isspth | Structured version Visualization version GIF version |
Description: Conditions for a pair of classes/functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.) |
Ref | Expression |
---|---|
isspth | ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spthsfval 28670 | . 2 ⊢ (SPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)} | |
2 | cnveq 5829 | . . . 4 ⊢ (𝑝 = 𝑃 → ◡𝑝 = ◡𝑃) | |
3 | 2 | funeqd 6523 | . . 3 ⊢ (𝑝 = 𝑃 → (Fun ◡𝑝 ↔ Fun ◡𝑃)) |
4 | 3 | adantl 482 | . 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ◡𝑝 ↔ Fun ◡𝑃)) |
5 | reltrls 28642 | . 2 ⊢ Rel (Trails‘𝐺) | |
6 | 1, 4, 5 | brfvopabrbr 6945 | 1 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 class class class wbr 5105 ◡ccnv 5632 Fun wfun 6490 ‘cfv 6496 Trailsctrls 28638 SPathscspths 28661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fv 6504 df-trls 28640 df-spths 28665 |
This theorem is referenced by: spthispth 28674 spthdifv 28681 spthdep 28682 pthdepisspth 28683 spthonepeq 28700 uhgrwkspth 28703 usgr2wlkspth 28707 usgr2pth 28712 2spthd 28886 0spth 29070 3spthd 29120 spthcycl 33723 |
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