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Mirrors > Home > MPE Home > Th. List > 1trld | Structured version Visualization version GIF version |
Description: In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
1wlkd.p | ⊢ 𝑃 = 〈“𝑋𝑌”〉 |
1wlkd.f | ⊢ 𝐹 = 〈“𝐽”〉 |
1wlkd.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
1wlkd.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
1wlkd.l | ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) |
1wlkd.j | ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) |
1wlkd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
1wlkd.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
1trld | ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1wlkd.p | . . 3 ⊢ 𝑃 = 〈“𝑋𝑌”〉 | |
2 | 1wlkd.f | . . 3 ⊢ 𝐹 = 〈“𝐽”〉 | |
3 | 1wlkd.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
4 | 1wlkd.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
5 | 1wlkd.l | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) | |
6 | 1wlkd.j | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) | |
7 | 1wlkd.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | 1wlkd.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | 1wlkd 27476 | . 2 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
10 | funcnvs1 13993 | . . 3 ⊢ Fun ◡〈“𝐽”〉 | |
11 | 2 | cnveqi 5498 | . . . 4 ⊢ ◡𝐹 = ◡〈“𝐽”〉 |
12 | 11 | funeqi 6120 | . . 3 ⊢ (Fun ◡𝐹 ↔ Fun ◡〈“𝐽”〉) |
13 | 10, 12 | mpbir 223 | . 2 ⊢ Fun ◡𝐹 |
14 | istrl 26940 | . 2 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹)) | |
15 | 9, 13, 14 | sylanblrc 585 | 1 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 ⊆ wss 3767 {csn 4366 {cpr 4368 class class class wbr 4841 ◡ccnv 5309 Fun wfun 6093 ‘cfv 6099 〈“cs1 13611 〈“cs2 13922 Vtxcvtx 26222 iEdgciedg 26223 Walkscwlks 26837 Trailsctrls 26934 |
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 2375 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
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 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-map 8095 df-pm 8096 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-card 9049 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-n0 11577 df-z 11663 df-uz 11927 df-fz 12577 df-fzo 12717 df-hash 13367 df-word 13531 df-concat 13587 df-s1 13612 df-s2 13929 df-wlks 26840 df-trls 26936 |
This theorem is referenced by: 1pthd 27478 1pthond 27479 upgr1trld 27483 |
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