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Mirrors > Home > MPE Home > Th. List > Mathboxes > upgrspths1wlk | Structured version Visualization version GIF version |
Description: The set of simple paths in a pseudograph, expressed as walk. Notice that this theorem would not hold for arbitrary hypergraphs, since a walk with distinct vertices does not need to be a trail: let E = { p0, p1, p2 } be a hyperedge, then ( p0, e, p1, e, p2 ) is walk with distinct vertices, but not with distinct edges. Therefore, E is not a trail and, by definition, also no path. (Contributed by AV, 11-Jan-2021.) (Proof shortened by AV, 17-Jan-2021.) |
Ref | Expression |
---|---|
upgrspths1wlk | ⊢ (𝐺 ∈ UPGraph → (SPathS‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spthsfval 40928 | . 2 ⊢ (𝐺 ∈ UPGraph → (SPathS‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(TrailS‘𝐺)𝑝 ∧ Fun ◡𝑝)}) | |
2 | vex 3176 | . . . . . . . 8 ⊢ 𝑓 ∈ V | |
3 | vex 3176 | . . . . . . . 8 ⊢ 𝑝 ∈ V | |
4 | isTrl 40904 | . . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑓 ∈ V ∧ 𝑝 ∈ V) → (𝑓(TrailS‘𝐺)𝑝 ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun ◡𝑓))) | |
5 | 2, 3, 4 | mp3an23 1408 | . . . . . . 7 ⊢ (𝐺 ∈ UPGraph → (𝑓(TrailS‘𝐺)𝑝 ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun ◡𝑓))) |
6 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ Fun ◡𝑝) → (𝑓(TrailS‘𝐺)𝑝 ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun ◡𝑓))) |
7 | upgrwlkdvde 40943 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑓(1Walks‘𝐺)𝑝 ∧ Fun ◡𝑝) → Fun ◡𝑓) | |
8 | 7 | 3exp 1256 | . . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (𝑓(1Walks‘𝐺)𝑝 → (Fun ◡𝑝 → Fun ◡𝑓))) |
9 | 8 | com23 84 | . . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (Fun ◡𝑝 → (𝑓(1Walks‘𝐺)𝑝 → Fun ◡𝑓))) |
10 | 9 | imp 444 | . . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ Fun ◡𝑝) → (𝑓(1Walks‘𝐺)𝑝 → Fun ◡𝑓)) |
11 | 10 | pm4.71d 664 | . . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ Fun ◡𝑝) → (𝑓(1Walks‘𝐺)𝑝 ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun ◡𝑓))) |
12 | 6, 11 | bitr4d 270 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ Fun ◡𝑝) → (𝑓(TrailS‘𝐺)𝑝 ↔ 𝑓(1Walks‘𝐺)𝑝)) |
13 | 12 | ex 449 | . . . 4 ⊢ (𝐺 ∈ UPGraph → (Fun ◡𝑝 → (𝑓(TrailS‘𝐺)𝑝 ↔ 𝑓(1Walks‘𝐺)𝑝))) |
14 | 13 | pm5.32rd 670 | . . 3 ⊢ (𝐺 ∈ UPGraph → ((𝑓(TrailS‘𝐺)𝑝 ∧ Fun ◡𝑝) ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun ◡𝑝))) |
15 | 14 | opabbidv 4648 | . 2 ⊢ (𝐺 ∈ UPGraph → {〈𝑓, 𝑝〉 ∣ (𝑓(TrailS‘𝐺)𝑝 ∧ Fun ◡𝑝)} = {〈𝑓, 𝑝〉 ∣ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)}) |
16 | 1, 15 | eqtrd 2644 | 1 ⊢ (𝐺 ∈ UPGraph → (SPathS‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 {copab 4642 ◡ccnv 5037 Fun wfun 5798 ‘cfv 5804 UPGraph cupgr 25747 1Walksc1wlks 40796 TrailSctrls 40899 SPathScspths 40920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-uhgr 25724 df-upgr 25749 df-edga 25793 df-1wlks 40800 df-wlks 40801 df-trls 40901 df-spths 40924 |
This theorem is referenced by: upgrwlkdvspth 40945 |
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