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Mirrors > Home > MPE Home > Th. List > trlres | Structured version Visualization version GIF version |
Description: The restriction 〈𝐻, 𝑄〉 of a trail 〈𝐹, 𝑃〉 to an initial segment of the trail (of length 𝑁) forms a trail on the subgraph 𝑆 consisting of the edges in the initial segment. (Contributed by AV, 6-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
Ref | Expression |
---|---|
trlres.v | ⊢ 𝑉 = (Vtx‘𝐺) |
trlres.i | ⊢ 𝐼 = (iEdg‘𝐺) |
trlres.d | ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
trlres.n | ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) |
trlres.h | ⊢ 𝐻 = (𝐹 prefix 𝑁) |
trlres.s | ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
trlres.e | ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
trlres.q | ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) |
Ref | Expression |
---|---|
trlres | ⊢ (𝜑 → 𝐻(Trails‘𝑆)𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlres.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | trlres.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | trlres.d | . . . 4 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | |
4 | trliswlk 28944 | . . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
6 | trlres.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) | |
7 | trlres.s | . . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | |
8 | trlres.e | . . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | |
9 | trlres.h | . . 3 ⊢ 𝐻 = (𝐹 prefix 𝑁) | |
10 | trlres.q | . . 3 ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) | |
11 | 1, 2, 5, 6, 7, 8, 9, 10 | wlkres 28917 | . 2 ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄) |
12 | 1, 2, 3, 6, 9 | trlreslem 28946 | . . 3 ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
13 | f1of1 6830 | . . 3 ⊢ (𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) → 𝐻:(0..^(♯‘𝐻))–1-1→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | |
14 | df-f1 6546 | . . . 4 ⊢ (𝐻:(0..^(♯‘𝐻))–1-1→dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) ↔ (𝐻:(0..^(♯‘𝐻))⟶dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) ∧ Fun ◡𝐻)) | |
15 | 14 | simprbi 498 | . . 3 ⊢ (𝐻:(0..^(♯‘𝐻))–1-1→dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) → Fun ◡𝐻) |
16 | 12, 13, 15 | 3syl 18 | . 2 ⊢ (𝜑 → Fun ◡𝐻) |
17 | istrl 28943 | . 2 ⊢ (𝐻(Trails‘𝑆)𝑄 ↔ (𝐻(Walks‘𝑆)𝑄 ∧ Fun ◡𝐻)) | |
18 | 11, 16, 17 | sylanbrc 584 | 1 ⊢ (𝜑 → 𝐻(Trails‘𝑆)𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 class class class wbr 5148 ◡ccnv 5675 dom cdm 5676 ↾ cres 5678 “ cima 5679 Fun wfun 6535 ⟶wf 6537 –1-1→wf1 6538 –1-1-onto→wf1o 6540 ‘cfv 6541 (class class class)co 7406 0cc0 11107 ...cfz 13481 ..^cfzo 13624 ♯chash 14287 prefix cpfx 14617 Vtxcvtx 28246 iEdgciedg 28247 Walkscwlks 28843 Trailsctrls 28937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 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-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 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625 df-hash 14288 df-word 14462 df-substr 14588 df-pfx 14618 df-wlks 28846 df-trls 28939 |
This theorem is referenced by: (None) |
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