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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 27406 | . . . 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 27379 | . 2 ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄) |
12 | 1, 2, 3, 6, 9 | trlreslem 27408 | . . 3 ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
13 | f1of1 6607 | . . 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 6353 | . . . 4 ⊢ (𝐻:(0..^(♯‘𝐻))–1-1→dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) ↔ (𝐻:(0..^(♯‘𝐻))⟶dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) ∧ Fun ◡𝐻)) | |
15 | 14 | simprbi 497 | . . 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 27405 | . 2 ⊢ (𝐻(Trails‘𝑆)𝑄 ↔ (𝐻(Walks‘𝑆)𝑄 ∧ Fun ◡𝐻)) | |
18 | 11, 16, 17 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐻(Trails‘𝑆)𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ◡ccnv 5547 dom cdm 5548 ↾ cres 5550 “ cima 5551 Fun wfun 6342 ⟶wf 6344 –1-1→wf1 6345 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 0cc0 10525 ...cfz 12880 ..^cfzo 13021 ♯chash 13678 prefix cpfx 14020 Vtxcvtx 26708 iEdgciedg 26709 Walkscwlks 27305 Trailsctrls 27399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-substr 13991 df-pfx 14021 df-wlks 27308 df-trls 27401 |
This theorem is referenced by: (None) |
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