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Mirrors > Home > MPE Home > Th. List > trlsegvdeglem2 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 28006. (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 6397 | . . 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 6377 | . 2 ⊢ (𝜑 → (Fun (iEdg‘𝑋) ↔ Fun (𝐼 ↾ (𝐹 “ (0..^𝑁))))) |
6 | 3, 5 | mpbird 259 | 1 ⊢ (𝜑 → Fun (iEdg‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {csn 4567 〈cop 4573 class class class wbr 5066 ↾ cres 5557 “ cima 5558 Fun wfun 6349 ‘cfv 6355 (class class class)co 7156 0cc0 10537 ...cfz 12893 ..^cfzo 13034 ♯chash 13691 Vtxcvtx 26781 iEdgciedg 26782 Trailsctrls 27472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3943 df-ss 3952 df-br 5067 df-opab 5129 df-rel 5562 df-cnv 5563 df-co 5564 df-res 5567 df-fun 6357 |
This theorem is referenced by: trlsegvdeg 28006 |
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