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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlsegvdeglem2 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 41395. (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 | funres 5843 | . . 3 ⊢ (Fun 𝐼 → Fun (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → Fun (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
4 | trlsegvdeg.ix | . . 3 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | |
5 | 4 | funeqd 5825 | . 2 ⊢ (𝜑 → (Fun (iEdg‘𝑋) ↔ Fun (𝐼 ↾ (𝐹 “ (0..^𝑁))))) |
6 | 3, 5 | mpbird 246 | 1 ⊢ (𝜑 → Fun (iEdg‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {csn 4125 〈cop 4131 class class class wbr 4583 ↾ cres 5040 “ cima 5041 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ...cfz 12197 ..^cfzo 12334 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 TrailSctrls 40899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-res 5050 df-fun 5806 |
This theorem is referenced by: trlsegvdeg 41395 |
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