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Mirrors > Home > MPE Home > Th. List > trlsegvdeglem2 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 28282. (Contributed by AV, 20-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 |
---|---|
trlsegvdeglem2 | ⊢ (𝜑 → Fun (iEdg‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlsegvdeg.f | . . 3 ⊢ (𝜑 → Fun 𝐼) | |
2 | 1 | funresd 6412 | . 2 ⊢ (𝜑 → Fun (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
3 | trlsegvdeg.ix | . . 3 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | |
4 | 3 | funeqd 6391 | . 2 ⊢ (𝜑 → (Fun (iEdg‘𝑋) ↔ Fun (𝐼 ↾ (𝐹 “ (0..^𝑁))))) |
5 | 2, 4 | mpbird 260 | 1 ⊢ (𝜑 → Fun (iEdg‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 {csn 4531 〈cop 4537 class class class wbr 5043 ↾ cres 5542 “ cima 5543 Fun wfun 6363 ‘cfv 6369 (class class class)co 7202 0cc0 10712 ...cfz 13078 ..^cfzo 13221 ♯chash 13879 Vtxcvtx 27059 iEdgciedg 27060 Trailsctrls 27750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-v 3403 df-in 3864 df-ss 3874 df-br 5044 df-opab 5106 df-rel 5547 df-cnv 5548 df-co 5549 df-res 5552 df-fun 6371 |
This theorem is referenced by: trlsegvdeg 28282 |
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