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Mirrors > Home > MPE Home > Th. List > trlsegvdeglem7 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 30256. (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 |
---|---|
trlsegvdeglem7 | ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ 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 | trlsegvdeglem5 30253 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)}) |
14 | snfi 9082 | . 2 ⊢ {(𝐹‘𝑁)} ∈ Fin | |
15 | 13, 14 | eqeltrdi 2847 | 1 ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {csn 4631 〈cop 4637 class class class wbr 5148 dom cdm 5689 ↾ cres 5691 “ cima 5692 Fun wfun 6557 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 0cc0 11153 ...cfz 13544 ..^cfzo 13691 ♯chash 14366 Vtxcvtx 29028 iEdgciedg 29029 Trailsctrls 29723 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1o 8505 df-en 8985 df-fin 8988 |
This theorem is referenced by: trlsegvdeg 30256 eupth2lem3lem2 30258 |
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