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| 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 | 
|---|---|
| trlsegvdeglem7 | ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ Fin) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | trlsegvdeg.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 2 | trlsegvdeg.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
| 3 | trlsegvdeg.f | . . 3 ⊢ (𝜑 → Fun 𝐼) | |
| 4 | trlsegvdeg.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) | |
| 5 | trlsegvdeg.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
| 6 | trlsegvdeg.w | . . 3 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | |
| 7 | trlsegvdeg.vx | . . 3 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) | |
| 8 | trlsegvdeg.vy | . . 3 ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) | |
| 9 | trlsegvdeg.vz | . . 3 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) | |
| 10 | trlsegvdeg.ix | . . 3 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | |
| 11 | trlsegvdeg.iy | . . 3 ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | |
| 12 | trlsegvdeg.iz | . . 3 ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) | |
| 13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | trlsegvdeglem5 30243 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)}) | 
| 14 | snfi 9083 | . 2 ⊢ {(𝐹‘𝑁)} ∈ Fin | |
| 15 | 13, 14 | eqeltrdi 2849 | 1 ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ Fin) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {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 Fincfn 8985 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-12 2177 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-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-en 8986 df-fin 8989 | 
| This theorem is referenced by: trlsegvdeg 30246 eupth2lem3lem2 30248 | 
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