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Mirrors > Home > MPE Home > Th. List > trlsegvdeglem6 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 28000. (Contributed by AV, 21-Feb-2021.) |
Ref | Expression |
---|---|
trlsegvdeg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
trlsegvdeg.i | ⊢ 𝐼 = (iEdg‘𝐺) |
trlsegvdeg.f | ⊢ (𝜑 → Fun 𝐼) |
trlsegvdeg.n | ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) |
trlsegvdeg.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
trlsegvdeg.w | ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
trlsegvdeg.vx | ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) |
trlsegvdeg.vy | ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) |
trlsegvdeg.vz | ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) |
trlsegvdeg.ix | ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
trlsegvdeg.iy | ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) |
trlsegvdeg.iz | ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) |
Ref | Expression |
---|---|
trlsegvdeglem6 | ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlsegvdeg.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | trlsegvdeg.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | trlsegvdeg.f | . . 3 ⊢ (𝜑 → Fun 𝐼) | |
4 | trlsegvdeg.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) | |
5 | trlsegvdeg.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
6 | trlsegvdeg.w | . . 3 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | |
7 | trlsegvdeg.vx | . . 3 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) | |
8 | trlsegvdeg.vy | . . 3 ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) | |
9 | trlsegvdeg.vz | . . 3 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) | |
10 | trlsegvdeg.ix | . . 3 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | |
11 | trlsegvdeg.iy | . . 3 ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | |
12 | trlsegvdeg.iz | . . 3 ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | trlsegvdeglem4 27996 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)) |
14 | 2 | trlf1 27474 | . . . . 5 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼) |
15 | f1fun 6572 | . . . . 5 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → Fun 𝐹) | |
16 | 6, 14, 15 | 3syl 18 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
17 | fzofi 13336 | . . . 4 ⊢ (0..^𝑁) ∈ Fin | |
18 | imafi 8811 | . . . 4 ⊢ ((Fun 𝐹 ∧ (0..^𝑁) ∈ Fin) → (𝐹 “ (0..^𝑁)) ∈ Fin) | |
19 | 16, 17, 18 | sylancl 588 | . . 3 ⊢ (𝜑 → (𝐹 “ (0..^𝑁)) ∈ Fin) |
20 | infi 8736 | . . 3 ⊢ ((𝐹 “ (0..^𝑁)) ∈ Fin → ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∈ Fin) | |
21 | 19, 20 | syl 17 | . 2 ⊢ (𝜑 → ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∈ Fin) |
22 | 13, 21 | eqeltrd 2913 | 1 ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∩ cin 3935 {csn 4561 〈cop 4567 class class class wbr 5059 dom cdm 5550 ↾ cres 5552 “ cima 5553 Fun wfun 6344 –1-1→wf1 6347 ‘cfv 6350 (class class class)co 7150 Fincfn 8503 0cc0 10531 ...cfz 12886 ..^cfzo 13027 ♯chash 13684 Vtxcvtx 26775 iEdgciedg 26776 Trailsctrls 27466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-wlks 27375 df-trls 27468 |
This theorem is referenced by: trlsegvdeg 28000 eupth2lem3lem1 28001 |
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