vtxdgfusgr - Metamath Proof Explorer

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Theorem vtxdgfusgr 29264

Description: In a finite simple graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 12-Dec-2020.)

**Hypothesis**
Ref	Expression
vtxdgfusgrf.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
vtxdgfusgr	⊢ (𝐺 ∈ FinUSGraph → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ₀)

Distinct variable group: 𝑣,𝐺 Allowed substitution hint: 𝑉(𝑣)

**Proof of Theorem vtxdgfusgr**
Step	Hyp	Ref	Expression
1		vtxdgfusgrf.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	vtxdgfusgrf 29263	. . 3 ⊢ (𝐺 ∈ FinUSGraph → (VtxDeg‘𝐺):𝑉⟶ℕ₀)
3	2	ffvelcdmda 7080	. 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ₀)
4	3	ralrimiva 3140	1 ⊢ (𝐺 ∈ FinUSGraph → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ₀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ‘cfv 6537 ℕ₀cn0 12476 Vtxcvtx 28764 FinUSGraphcfusgr 29081 VtxDegcvtxdg 29231

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-xadd 13099 df-fz 13491 df-hash 14296 df-vtx 28766 df-iedg 28767 df-edg 28816 df-uhgr 28826 df-upgr 28850 df-umgr 28851 df-uspgr 28918 df-usgr 28919 df-fusgr 29082 df-vtxdg 29232

This theorem is referenced by: fusgrn0degnn0 29265 fusgrregdegfi 29335

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