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Mirrors > Home > MPE Home > Th. List > 2trld | Structured version Visualization version GIF version |
Description: Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
2wlkd.p | ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 |
2wlkd.f | ⊢ 𝐹 = 〈“𝐽𝐾”〉 |
2wlkd.s | ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
2wlkd.n | ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) |
2wlkd.e | ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) |
2wlkd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
2wlkd.i | ⊢ 𝐼 = (iEdg‘𝐺) |
2trld.n | ⊢ (𝜑 → 𝐽 ≠ 𝐾) |
Ref | Expression |
---|---|
2trld | ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2wlkd.p | . . 3 ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 | |
2 | 2wlkd.f | . . 3 ⊢ 𝐹 = 〈“𝐽𝐾”〉 | |
3 | 2wlkd.s | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | |
4 | 2wlkd.n | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) | |
5 | 2wlkd.e | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) | |
6 | 2wlkd.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
7 | 2wlkd.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
8 | 1, 2, 3, 4, 5, 6, 7 | 2wlkd 27056 | . 2 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
9 | 1, 2, 3, 4, 5 | 2wlkdlem7 27052 | . . . . 5 ⊢ (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V)) |
10 | 2trld.n | . . . . 5 ⊢ (𝜑 → 𝐽 ≠ 𝐾) | |
11 | df-3an 1074 | . . . . 5 ⊢ ((𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐽 ≠ 𝐾) ↔ ((𝐽 ∈ V ∧ 𝐾 ∈ V) ∧ 𝐽 ≠ 𝐾)) | |
12 | 9, 10, 11 | sylanbrc 701 | . . . 4 ⊢ (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐽 ≠ 𝐾)) |
13 | funcnvs2 13858 | . . . 4 ⊢ ((𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐽 ≠ 𝐾) → Fun ◡〈“𝐽𝐾”〉) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → Fun ◡〈“𝐽𝐾”〉) |
15 | 2 | cnveqi 5452 | . . . 4 ⊢ ◡𝐹 = ◡〈“𝐽𝐾”〉 |
16 | 15 | funeqi 6070 | . . 3 ⊢ (Fun ◡𝐹 ↔ Fun ◡〈“𝐽𝐾”〉) |
17 | 14, 16 | sylibr 224 | . 2 ⊢ (𝜑 → Fun ◡𝐹) |
18 | istrl 26803 | . 2 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹)) | |
19 | 8, 17, 18 | sylanbrc 701 | 1 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 Vcvv 3340 ⊆ wss 3715 {cpr 4323 class class class wbr 4804 ◡ccnv 5265 Fun wfun 6043 ‘cfv 6049 〈“cs2 13786 〈“cs3 13787 Vtxcvtx 26073 iEdgciedg 26074 Walkscwlks 26702 Trailsctrls 26797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ifp 1051 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-map 8025 df-pm 8026 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-fzo 12660 df-hash 13312 df-word 13485 df-concat 13487 df-s1 13488 df-s2 13793 df-s3 13794 df-wlks 26705 df-trls 26799 |
This theorem is referenced by: 2trlond 27059 2pthd 27060 2spthd 27061 |
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