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Mirrors > Home > MPE Home > Th. List > 1pthond | 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 path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-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 |
---|---|
1pthond | ⊢ (𝜑 → 𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1wlkd.p | . . . . 5 ⊢ 𝑃 = 〈“𝑋𝑌”〉 | |
2 | 1wlkd.f | . . . . 5 ⊢ 𝐹 = 〈“𝐽”〉 | |
3 | 1wlkd.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
4 | 1wlkd.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
5 | 1wlkd.l | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) | |
6 | 1wlkd.j | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) | |
7 | 1wlkd.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | 1wlkd.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | 1wlkd 27518 | . . . 4 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
10 | 1 | fveq1i 6435 | . . . . . 6 ⊢ (𝑃‘0) = (〈“𝑋𝑌”〉‘0) |
11 | s2fv0 14009 | . . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (〈“𝑋𝑌”〉‘0) = 𝑋) | |
12 | 10, 11 | syl5eq 2874 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑃‘0) = 𝑋) |
13 | 3, 12 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑃‘0) = 𝑋) |
14 | 2 | fveq2i 6437 | . . . . . . 7 ⊢ (♯‘𝐹) = (♯‘〈“𝐽”〉) |
15 | s1len 13667 | . . . . . . 7 ⊢ (♯‘〈“𝐽”〉) = 1 | |
16 | 14, 15 | eqtri 2850 | . . . . . 6 ⊢ (♯‘𝐹) = 1 |
17 | 1, 16 | fveq12i 6440 | . . . . 5 ⊢ (𝑃‘(♯‘𝐹)) = (〈“𝑋𝑌”〉‘1) |
18 | s2fv1 14010 | . . . . . 6 ⊢ (𝑌 ∈ 𝑉 → (〈“𝑋𝑌”〉‘1) = 𝑌) | |
19 | 4, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → (〈“𝑋𝑌”〉‘1) = 𝑌) |
20 | 17, 19 | syl5eq 2874 | . . . 4 ⊢ (𝜑 → (𝑃‘(♯‘𝐹)) = 𝑌) |
21 | wlkv 26911 | . . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
22 | 3simpc 1188 | . . . . . . 7 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
23 | 9, 21, 22 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝑃 ∈ V)) |
24 | 3, 4, 23 | jca31 512 | . . . . 5 ⊢ (𝜑 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
25 | 7 | iswlkon 26955 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑋(WalksOn‘𝐺)𝑌)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(♯‘𝐹)) = 𝑌))) |
26 | 24, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐹(𝑋(WalksOn‘𝐺)𝑌)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(♯‘𝐹)) = 𝑌))) |
27 | 9, 13, 20, 26 | mpbir3and 1448 | . . 3 ⊢ (𝜑 → 𝐹(𝑋(WalksOn‘𝐺)𝑌)𝑃) |
28 | 1, 2, 3, 4, 5, 6, 7, 8 | 1trld 27519 | . . 3 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
29 | 7 | istrlson 27010 | . . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑋(TrailsOn‘𝐺)𝑌)𝑃 ↔ (𝐹(𝑋(WalksOn‘𝐺)𝑌)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃))) |
30 | 24, 29 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹(𝑋(TrailsOn‘𝐺)𝑌)𝑃 ↔ (𝐹(𝑋(WalksOn‘𝐺)𝑌)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃))) |
31 | 27, 28, 30 | mpbir2and 706 | . 2 ⊢ (𝜑 → 𝐹(𝑋(TrailsOn‘𝐺)𝑌)𝑃) |
32 | 1, 2, 3, 4, 5, 6, 7, 8 | 1pthd 27520 | . 2 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
33 | 3 | adantl 475 | . . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → 𝑋 ∈ 𝑉) |
34 | 4 | adantl 475 | . . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → 𝑌 ∈ 𝑉) |
35 | simpl 476 | . . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → (𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
36 | 33, 34, 35 | jca31 512 | . . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
37 | 36 | ex 403 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝜑 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))) |
38 | 21, 22, 37 | 3syl 18 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝜑 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))) |
39 | 9, 38 | mpcom 38 | . . 3 ⊢ (𝜑 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
40 | 7 | ispthson 27045 | . . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃 ↔ (𝐹(𝑋(TrailsOn‘𝐺)𝑌)𝑃 ∧ 𝐹(Paths‘𝐺)𝑃))) |
41 | 39, 40 | syl 17 | . 2 ⊢ (𝜑 → (𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃 ↔ (𝐹(𝑋(TrailsOn‘𝐺)𝑌)𝑃 ∧ 𝐹(Paths‘𝐺)𝑃))) |
42 | 31, 32, 41 | mpbir2and 706 | 1 ⊢ (𝜑 → 𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 Vcvv 3415 ⊆ wss 3799 {csn 4398 {cpr 4400 class class class wbr 4874 ‘cfv 6124 (class class class)co 6906 0cc0 10253 1c1 10254 ♯chash 13411 〈“cs1 13656 〈“cs2 13963 Vtxcvtx 26295 iEdgciedg 26296 Walkscwlks 26895 WalksOncwlkson 26896 Trailsctrls 26992 TrailsOnctrlson 26993 Pathscpths 27015 PathsOncpthson 27017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-ifp 1092 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-map 8125 df-pm 8126 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-card 9079 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-n0 11620 df-z 11706 df-uz 11970 df-fz 12621 df-fzo 12762 df-hash 13412 df-word 13576 df-concat 13632 df-s1 13657 df-s2 13970 df-wlks 26898 df-wlkson 26899 df-trls 26994 df-trlson 26995 df-pths 27019 df-pthson 27021 |
This theorem is referenced by: upgr1pthond 27527 lppthon 27528 1pthon2v 27530 |
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