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Mirrors > Home > MPE Home > Th. List > 1pthd | 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. 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, 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 |
---|---|
1pthd | ⊢ (𝜑 → 𝐹(Paths‘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 | 1trld 28081 | . 2 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
10 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝐹(Trails‘𝐺)𝑃) → 𝐹(Trails‘𝐺)𝑃) | |
11 | 1, 2 | 1pthdlem1 28074 | . . . 4 ⊢ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) |
12 | 11 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝐹(Trails‘𝐺)𝑃) → Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))) |
13 | 1, 2 | 1pthdlem2 28075 | . . . 4 ⊢ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅ |
14 | 13 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝐹(Trails‘𝐺)𝑃) → ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅) |
15 | ispth 27666 | . . 3 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅)) | |
16 | 10, 12, 14, 15 | syl3anbrc 1344 | . 2 ⊢ ((𝜑 ∧ 𝐹(Trails‘𝐺)𝑃) → 𝐹(Paths‘𝐺)𝑃) |
17 | 9, 16 | mpdan 687 | 1 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 {csn 4516 {cpr 4518 class class class wbr 5030 ◡ccnv 5524 ↾ cres 5527 “ cima 5528 Fun wfun 6333 ‘cfv 6339 (class class class)co 7172 0cc0 10617 1c1 10618 ..^cfzo 13126 ♯chash 13784 〈“cs1 14040 〈“cs2 14294 Vtxcvtx 26943 iEdgciedg 26944 Trailsctrls 27634 Pathscpths 27655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-map 8441 df-pm 8442 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-card 9443 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-n0 11979 df-z 12065 df-uz 12327 df-fz 12984 df-fzo 13127 df-hash 13785 df-word 13958 df-concat 14014 df-s1 14041 df-s2 14301 df-wlks 27543 df-trls 27636 df-pths 27659 |
This theorem is referenced by: 1pthond 28083 upgr1pthd 28088 |
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