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Mirrors > Home > MPE Home > Th. List > trlsegvdeglem2 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 29345. (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 6580 | . 2 ⊢ (𝜑 → Fun (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
3 | trlsegvdeg.ix | . . 3 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | |
4 | 3 | funeqd 6559 | . 2 ⊢ (𝜑 → (Fun (iEdg‘𝑋) ↔ Fun (𝐼 ↾ (𝐹 “ (0..^𝑁))))) |
5 | 2, 4 | mpbird 256 | 1 ⊢ (𝜑 → Fun (iEdg‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {csn 4622 〈cop 4628 class class class wbr 5141 ↾ cres 5671 “ cima 5672 Fun wfun 6526 ‘cfv 6532 (class class class)co 7393 0cc0 11092 ...cfz 13466 ..^cfzo 13609 ♯chash 14272 Vtxcvtx 28121 iEdgciedg 28122 Trailsctrls 28812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-in 3951 df-ss 3961 df-br 5142 df-opab 5204 df-rel 5676 df-cnv 5677 df-co 5678 df-res 5681 df-fun 6534 |
This theorem is referenced by: trlsegvdeg 29345 |
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