| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > trlsegvdeglem5 | Structured version Visualization version GIF version | ||
| Description: Lemma for trlsegvdeg 30246. (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 5916 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑌) = dom {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) |
| 3 | fvex 6919 | . . 3 ⊢ (𝐼‘(𝐹‘𝑁)) ∈ V | |
| 4 | dmsnopg 6233 | . . 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 2777 | 1 ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 class class class wbr 5143 dom cdm 5685 ↾ cres 5687 “ cima 5688 Fun wfun 6555 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ...cfz 13547 ..^cfzo 13694 ♯chash 14369 Vtxcvtx 29013 iEdgciedg 29014 Trailsctrls 29708 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: trlsegvdeglem7 30245 trlsegvdeg 30246 |
| Copyright terms: Public domain | W3C validator |