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Mirrors > Home > MPE Home > Th. List > trlsegvdeglem2 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 27933. (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 6390 | . . 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 6370 | . 2 ⊢ (𝜑 → (Fun (iEdg‘𝑋) ↔ Fun (𝐼 ↾ (𝐹 “ (0..^𝑁))))) |
6 | 3, 5 | mpbird 258 | 1 ⊢ (𝜑 → Fun (iEdg‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {csn 4557 〈cop 4563 class class class wbr 5057 ↾ cres 5550 “ cima 5551 Fun wfun 6342 ‘cfv 6348 (class class class)co 7145 0cc0 10525 ...cfz 12880 ..^cfzo 13021 ♯chash 13678 Vtxcvtx 26708 iEdgciedg 26709 Trailsctrls 27399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-in 3940 df-ss 3949 df-br 5058 df-opab 5120 df-rel 5555 df-cnv 5556 df-co 5557 df-res 5560 df-fun 6350 |
This theorem is referenced by: trlsegvdeg 27933 |
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