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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 | 
|---|---|
| trlsegvdeglem6 | ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ 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 | trlsegvdeglem4 30242 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)) | 
| 14 | 2 | trlf1 29716 | . . . . 5 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼) | 
| 15 | f1fun 6806 | . . . . 5 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → Fun 𝐹) | |
| 16 | 6, 14, 15 | 3syl 18 | . . . 4 ⊢ (𝜑 → Fun 𝐹) | 
| 17 | fzofi 14015 | . . . 4 ⊢ (0..^𝑁) ∈ Fin | |
| 18 | imafi 9353 | . . . 4 ⊢ ((Fun 𝐹 ∧ (0..^𝑁) ∈ Fin) → (𝐹 “ (0..^𝑁)) ∈ Fin) | |
| 19 | 16, 17, 18 | sylancl 586 | . . 3 ⊢ (𝜑 → (𝐹 “ (0..^𝑁)) ∈ Fin) | 
| 20 | infi 9302 | . . 3 ⊢ ((𝐹 “ (0..^𝑁)) ∈ Fin → ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∈ Fin) | |
| 21 | 19, 20 | syl 17 | . 2 ⊢ (𝜑 → ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∈ Fin) | 
| 22 | 13, 21 | eqeltrd 2841 | 1 ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 {csn 4626 〈cop 4632 class class class wbr 5143 dom cdm 5685 ↾ cres 5687 “ cima 5688 Fun wfun 6555 –1-1→wf1 6558 ‘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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 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-res 5697 df-ima 5698 df-pred 6321 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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-wlks 29617 df-trls 29710 | 
| This theorem is referenced by: trlsegvdeg 30246 eupth2lem3lem1 30247 | 
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