![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 254 | . . 3 ⊢ ((⊤ ∧ 𝑔 = 𝐺) → (Fun ◡𝑝 ↔ Fun ◡𝑝)) | |
2 | wksv 26862 | . . . . 5 ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V | |
3 | trliswlk 26943 | . . . . . 6 ⊢ (𝑓(Trails‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝) | |
4 | 3 | ssopab2i 5198 | . . . . 5 ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Trails‘𝐺)𝑝} ⊆ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} |
5 | 2, 4 | ssexi 4997 | . . . 4 ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Trails‘𝐺)𝑝} ∈ V |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → {〈𝑓, 𝑝〉 ∣ 𝑓(Trails‘𝐺)𝑝} ∈ V) |
7 | df-spths 26964 | . . 3 ⊢ SPaths = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡𝑝)}) | |
8 | 1, 6, 7 | fvmptopab 6930 | . 2 ⊢ (⊤ → (SPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)}) |
9 | 8 | mptru 1661 | 1 ⊢ (SPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 385 = wceq 1653 ⊤wtru 1654 ∈ wcel 2157 Vcvv 3384 class class class wbr 4842 {copab 4904 ◡ccnv 5310 Fun wfun 6094 ‘cfv 6100 Walkscwlks 26839 Trailsctrls 26936 SPathscspths 26960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-rep 4963 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-ifp 1087 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-int 4667 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-om 7299 df-1st 7400 df-2nd 7401 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-1o 7798 df-er 7981 df-map 8096 df-pm 8097 df-en 8195 df-dom 8196 df-sdom 8197 df-fin 8198 df-card 9050 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-nn 11312 df-n0 11578 df-z 11664 df-uz 11928 df-fz 12578 df-fzo 12718 df-hash 13368 df-word 13532 df-wlks 26842 df-trls 26938 df-spths 26964 |
This theorem is referenced by: isspth 26971 upgrspthswlk 26985 |
Copyright terms: Public domain | W3C validator |