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Theorem eupth2lem3lem1 30257

Description: Lemma for eupth2lem3 30265. (Contributed by AV, 21-Feb-2021.)

**Hypotheses**
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))))

**Assertion**
Ref	Expression
eupth2lem3lem1	⊢ (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ₀)

**Proof of Theorem eupth2lem3lem1**
Step	Hyp	Ref	Expression
1		trlsegvdeg.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
2		trlsegvdeg.vx	. . . . 5 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)
3	1, 2	eleqtrrd 2842	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑋))
4	3	elfvexd 6946	. . 3 ⊢ (𝜑 → 𝑋 ∈ V)
5		trlsegvdeg.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
6		trlsegvdeg.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
7		trlsegvdeg.f	. . . 4 ⊢ (𝜑 → Fun 𝐼)
8		trlsegvdeg.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
9		trlsegvdeg.w	. . . 4 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)
10		trlsegvdeg.vy	. . . 4 ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)
11		trlsegvdeg.vz	. . . 4 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)
12		trlsegvdeg.ix	. . . 4 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
13		trlsegvdeg.iy	. . . 4 ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
14		trlsegvdeg.iz	. . . 4 ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
15	5, 6, 7, 8, 1, 9, 2, 10, 11, 12, 13, 14	trlsegvdeglem6 30254	. . 3 ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin)
16		eqid 2735	. . . 4 ⊢ (Vtx‘𝑋) = (Vtx‘𝑋)
17		eqid 2735	. . . 4 ⊢ (iEdg‘𝑋) = (iEdg‘𝑋)
18		eqid 2735	. . . 4 ⊢ dom (iEdg‘𝑋) = dom (iEdg‘𝑋)
19	16, 17, 18	vtxdgfisf 29509	. . 3 ⊢ ((𝑋 ∈ V ∧ dom (iEdg‘𝑋) ∈ Fin) → (VtxDeg‘𝑋):(Vtx‘𝑋)⟶ℕ₀)
20	4, 15, 19	syl2anc 584	. 2 ⊢ (𝜑 → (VtxDeg‘𝑋):(Vtx‘𝑋)⟶ℕ₀)
21	20, 3	ffvelcdmd 7105	1 ⊢ (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ₀)

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 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 0cc0 11153 ℕ₀cn0 12524 ...cfz 13544 ..^cfzo 13691 ♯chash 14366 Vtxcvtx 29028 iEdgciedg 29029 VtxDegcvtxdg 29498 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-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-xadd 13153 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-vtxdg 29499 df-wlks 29632 df-trls 29725

This theorem is referenced by: eupth2lem3lem3 30259 eupth2lem3lem4 30260 eupth2lem3lem6 30262

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