vtxdgfisf - Metamath Proof Explorer

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Theorem vtxdgfisf 27746

Description: The vertex degree function on graphs of finite size is a function from vertices to nonnegative integers. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.)

**Hypotheses**
Ref	Expression
vtxdgf.v	⊢ 𝑉 = (Vtx‘𝐺)
vtxdg0e.i	⊢ 𝐼 = (iEdg‘𝐺)
vtxdgfisnn0.a	⊢ 𝐴 = dom 𝐼

**Assertion**
Ref	Expression
vtxdgfisf	⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → (VtxDeg‘𝐺):𝑉⟶ℕ₀)

Proof of Theorem vtxdgfisf Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vtxdgf.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	vtxdgf 27741	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ₀^*)
3	2	adantr 480	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → (VtxDeg‘𝐺):𝑉⟶ℕ₀^*)
4	3	ffnd 6585	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → (VtxDeg‘𝐺) Fn 𝑉)
5		vtxdg0e.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
6		vtxdgfisnn0.a	. . . . 5 ⊢ 𝐴 = dom 𝐼
7	1, 5, 6	vtxdgfisnn0 27745	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑢 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀)
8	7	adantll 710	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) ∧ 𝑢 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀)
9	8	ralrimiva 3107	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → ∀𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀)
10		ffnfv 6974	. 2 ⊢ ((VtxDeg‘𝐺):𝑉⟶ℕ₀ ↔ ((VtxDeg‘𝐺) Fn 𝑉 ∧ ∀𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀))
11	4, 9, 10	sylanbrc 582	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → (VtxDeg‘𝐺):𝑉⟶ℕ₀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 dom cdm 5580 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 Fincfn 8691 ℕ₀cn0 12163 ℕ₀^*cxnn0 12235 Vtxcvtx 27269 iEdgciedg 27270 VtxDegcvtxdg 27735

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-xadd 12778 df-hash 13973 df-vtxdg 27736

This theorem is referenced by: vtxdgfusgrf 27767 eupth2lem3lem1 28493 eupth2lem3lem2 28494

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