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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 29740 | . 2 ⊢ (SPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)} | |
| 2 | cnveq 5884 | . . . 4 ⊢ (𝑝 = 𝑃 → ◡𝑝 = ◡𝑃) | |
| 3 | 2 | funeqd 6588 | . . 3 ⊢ (𝑝 = 𝑃 → (Fun ◡𝑝 ↔ Fun ◡𝑃)) |
| 4 | 3 | adantl 481 | . 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ◡𝑝 ↔ Fun ◡𝑃)) |
| 5 | reltrls 29712 | . 2 ⊢ Rel (Trails‘𝐺) | |
| 6 | 1, 4, 5 | brfvopabrbr 7013 | 1 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 class class class wbr 5143 ◡ccnv 5684 Fun wfun 6555 ‘cfv 6561 Trailsctrls 29708 SPathscspths 29731 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-trls 29710 df-spths 29735 |
| This theorem is referenced by: spthispth 29744 spthdifv 29753 spthdep 29754 pthdepisspth 29755 spthonepeq 29772 uhgrwkspth 29775 usgr2wlkspth 29779 usgr2pth 29784 2spthd 29961 0spth 30145 3spthd 30195 spthcycl 35134 |
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