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Mirrors > Home > MPE Home > Th. List > spthsfval | Structured version Visualization version GIF version |
Description: The set of simple paths (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 |
---|---|
spthsfval | ⊢ (SPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 261 | . 2 ⊢ (𝑔 = 𝐺 → (Fun ◡𝑝 ↔ Fun ◡𝑝)) | |
2 | df-spths 29654 | . 2 ⊢ SPaths = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡𝑝)}) | |
3 | 1, 2 | fvmptopab 7479 | 1 ⊢ (SPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1534 class class class wbr 5153 {copab 5215 ◡ccnv 5681 Fun wfun 6548 ‘cfv 6554 Trailsctrls 29627 SPathscspths 29650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6506 df-fun 6556 df-fv 6562 df-spths 29654 |
This theorem is referenced by: isspth 29661 upgrspthswlk 29675 |
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