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Mirrors > Home > MPE Home > Th. List > trlsegvdeglem5 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 30256. (Contributed by AV, 21-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 |
---|---|
trlsegvdeglem5 | ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlsegvdeg.iy | . . 3 ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | |
2 | 1 | dmeqd 5919 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑌) = dom {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) |
3 | fvex 6920 | . . 3 ⊢ (𝐼‘(𝐹‘𝑁)) ∈ V | |
4 | dmsnopg 6235 | . . 3 ⊢ ((𝐼‘(𝐹‘𝑁)) ∈ V → dom {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉} = {(𝐹‘𝑁)}) | |
5 | 3, 4 | mp1i 13 | . 2 ⊢ (𝜑 → dom {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉} = {(𝐹‘𝑁)}) |
6 | 2, 5 | eqtrd 2775 | 1 ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 〈cop 4637 class class class wbr 5148 dom cdm 5689 ↾ cres 5691 “ cima 5692 Fun wfun 6557 ‘cfv 6563 (class class class)co 7431 0cc0 11153 ...cfz 13544 ..^cfzo 13691 ♯chash 14366 Vtxcvtx 29028 iEdgciedg 29029 Trailsctrls 29723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-dm 5699 df-iota 6516 df-fv 6571 |
This theorem is referenced by: trlsegvdeglem7 30255 trlsegvdeg 30256 |
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