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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 30074 | . . . 4 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
10 | 1 | fveq1i 6902 | . . . . . 6 ⊢ (𝑃‘0) = (〈“𝑋𝑌”〉‘0) |
11 | s2fv0 14896 | . . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (〈“𝑋𝑌”〉‘0) = 𝑋) | |
12 | 10, 11 | eqtrid 2778 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑃‘0) = 𝑋) |
13 | 3, 12 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑃‘0) = 𝑋) |
14 | 2 | fveq2i 6904 | . . . . . . 7 ⊢ (♯‘𝐹) = (♯‘〈“𝐽”〉) |
15 | s1len 14614 | . . . . . . 7 ⊢ (♯‘〈“𝐽”〉) = 1 | |
16 | 14, 15 | eqtri 2754 | . . . . . 6 ⊢ (♯‘𝐹) = 1 |
17 | 1, 16 | fveq12i 6907 | . . . . 5 ⊢ (𝑃‘(♯‘𝐹)) = (〈“𝑋𝑌”〉‘1) |
18 | s2fv1 14897 | . . . . . 6 ⊢ (𝑌 ∈ 𝑉 → (〈“𝑋𝑌”〉‘1) = 𝑌) | |
19 | 4, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → (〈“𝑋𝑌”〉‘1) = 𝑌) |
20 | 17, 19 | eqtrid 2778 | . . . 4 ⊢ (𝜑 → (𝑃‘(♯‘𝐹)) = 𝑌) |
21 | wlkv 29549 | . . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
22 | 3simpc 1147 | . . . . . . 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 513 | . . . . 5 ⊢ (𝜑 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
25 | 7 | iswlkon 29594 | . . . . 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 1339 | . . 3 ⊢ (𝜑 → 𝐹(𝑋(WalksOn‘𝐺)𝑌)𝑃) |
28 | 1, 2, 3, 4, 5, 6, 7, 8 | 1trld 30075 | . . 3 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
29 | 7 | istrlson 29644 | . . . 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 711 | . 2 ⊢ (𝜑 → 𝐹(𝑋(TrailsOn‘𝐺)𝑌)𝑃) |
32 | 1, 2, 3, 4, 5, 6, 7, 8 | 1pthd 30076 | . 2 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
33 | 3 | adantl 480 | . . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → 𝑋 ∈ 𝑉) |
34 | 4 | adantl 480 | . . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → 𝑌 ∈ 𝑉) |
35 | simpl 481 | . . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → (𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
36 | 33, 34, 35 | jca31 513 | . . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝜑) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
37 | 36 | ex 411 | . . . . 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 29679 | . . 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 711 | 1 ⊢ (𝜑 → 𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 ⊆ wss 3947 {csn 4633 {cpr 4635 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 0cc0 11158 1c1 11159 ♯chash 14347 〈“cs1 14603 〈“cs2 14850 Vtxcvtx 28932 iEdgciedg 28933 Walkscwlks 29533 WalksOncwlkson 29534 Trailsctrls 29627 TrailsOnctrlson 29628 Pathscpths 29649 PathsOncpthson 29651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-fzo 13682 df-hash 14348 df-word 14523 df-concat 14579 df-s1 14604 df-s2 14857 df-wlks 29536 df-wlkson 29537 df-trls 29629 df-trlson 29630 df-pths 29653 df-pthson 29655 |
This theorem is referenced by: upgr1pthond 30083 lppthon 30084 1pthon2v 30086 |
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