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Mirrors > Home > MPE Home > Th. List > 2spthd | Structured version Visualization version GIF version |
Description: A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
2wlkd.p | β’ π = β¨βπ΄π΅πΆββ© |
2wlkd.f | β’ πΉ = β¨βπ½πΎββ© |
2wlkd.s | β’ (π β (π΄ β π β§ π΅ β π β§ πΆ β π)) |
2wlkd.n | β’ (π β (π΄ β π΅ β§ π΅ β πΆ)) |
2wlkd.e | β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ))) |
2wlkd.v | β’ π = (VtxβπΊ) |
2wlkd.i | β’ πΌ = (iEdgβπΊ) |
2trld.n | β’ (π β π½ β πΎ) |
2spthd.n | β’ (π β π΄ β πΆ) |
Ref | Expression |
---|---|
2spthd | β’ (π β πΉ(SPathsβπΊ)π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2wlkd.p | . . 3 β’ π = β¨βπ΄π΅πΆββ© | |
2 | 2wlkd.f | . . 3 β’ πΉ = β¨βπ½πΎββ© | |
3 | 2wlkd.s | . . 3 β’ (π β (π΄ β π β§ π΅ β π β§ πΆ β π)) | |
4 | 2wlkd.n | . . 3 β’ (π β (π΄ β π΅ β§ π΅ β πΆ)) | |
5 | 2wlkd.e | . . 3 β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ))) | |
6 | 2wlkd.v | . . 3 β’ π = (VtxβπΊ) | |
7 | 2wlkd.i | . . 3 β’ πΌ = (iEdgβπΊ) | |
8 | 2trld.n | . . 3 β’ (π β π½ β πΎ) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | 2trld 29447 | . 2 β’ (π β πΉ(TrailsβπΊ)π) |
10 | 2spthd.n | . . . . 5 β’ (π β π΄ β πΆ) | |
11 | 3anan32 1097 | . . . . 5 β’ ((π΄ β π΅ β§ π΄ β πΆ β§ π΅ β πΆ) β ((π΄ β π΅ β§ π΅ β πΆ) β§ π΄ β πΆ)) | |
12 | 4, 10, 11 | sylanbrc 583 | . . . 4 β’ (π β (π΄ β π΅ β§ π΄ β πΆ β§ π΅ β πΆ)) |
13 | funcnvs3 14869 | . . . 4 β’ (((π΄ β π β§ π΅ β π β§ πΆ β π) β§ (π΄ β π΅ β§ π΄ β πΆ β§ π΅ β πΆ)) β Fun β‘β¨βπ΄π΅πΆββ©) | |
14 | 3, 12, 13 | syl2anc 584 | . . 3 β’ (π β Fun β‘β¨βπ΄π΅πΆββ©) |
15 | 1 | a1i 11 | . . . . 5 β’ (π β π = β¨βπ΄π΅πΆββ©) |
16 | 15 | cnveqd 5875 | . . . 4 β’ (π β β‘π = β‘β¨βπ΄π΅πΆββ©) |
17 | 16 | funeqd 6570 | . . 3 β’ (π β (Fun β‘π β Fun β‘β¨βπ΄π΅πΆββ©)) |
18 | 14, 17 | mpbird 256 | . 2 β’ (π β Fun β‘π) |
19 | isspth 29236 | . 2 β’ (πΉ(SPathsβπΊ)π β (πΉ(TrailsβπΊ)π β§ Fun β‘π)) | |
20 | 9, 18, 19 | sylanbrc 583 | 1 β’ (π β πΉ(SPathsβπΊ)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 β wss 3948 {cpr 4630 class class class wbr 5148 β‘ccnv 5675 Fun wfun 6537 βcfv 6543 β¨βcs2 14796 β¨βcs3 14797 Vtxcvtx 28511 iEdgciedg 28512 Trailsctrls 29202 SPathscspths 29225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-hash 14295 df-word 14469 df-concat 14525 df-s1 14550 df-s2 14803 df-s3 14804 df-wlks 29111 df-trls 29204 df-spths 29229 |
This theorem is referenced by: 2pthond 29451 umgr2adedgspth 29457 |
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