vtxdgfisf - Metamath Proof Explorer

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Theorem vtxdgfisf 28132

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 28127	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ₀^*)
3	2	adantr 481	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → (VtxDeg‘𝐺):𝑉⟶ℕ₀^*)
4	3	ffnd 6652	. 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 28131	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑢 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀)
8	7	adantll 711	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) ∧ 𝑢 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀)
9	8	ralrimiva 3139	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → ∀𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀)
10		ffnfv 7048	. 2 ⊢ ((VtxDeg‘𝐺):𝑉⟶ℕ₀ ↔ ((VtxDeg‘𝐺) Fn 𝑉 ∧ ∀𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀))
11	4, 9, 10	sylanbrc 583	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → (VtxDeg‘𝐺):𝑉⟶ℕ₀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3061 dom cdm 5620 Fn wfn 6474 ⟶wf 6475 ‘cfv 6479 Fincfn 8804 ℕ₀cn0 12334 ℕ₀^*cxnn0 12406 Vtxcvtx 27655 iEdgciedg 27656 VtxDegcvtxdg 28121

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-n0 12335 df-xnn0 12407 df-z 12421 df-uz 12684 df-xadd 12950 df-hash 14146 df-vtxdg 28122

This theorem is referenced by: vtxdgfusgrf 28153 eupth2lem3lem1 28880 eupth2lem3lem2 28881

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