vtxdgfisf - Metamath Proof Explorer

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Theorem vtxdgfisf 29494

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 29489	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ₀^*)
3	2	adantr 480	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → (VtxDeg‘𝐺):𝑉⟶ℕ₀^*)
4	3	ffnd 6737	. 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 29493	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑢 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀)
8	7	adantll 714	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) ∧ 𝑢 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀)
9	8	ralrimiva 3146	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → ∀𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀)
10		ffnfv 7139	. 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 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 dom cdm 5685 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 Fincfn 8985 ℕ₀cn0 12526 ℕ₀^*cxnn0 12599 Vtxcvtx 29013 iEdgciedg 29014 VtxDegcvtxdg 29483

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-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-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 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-xnn0 12600 df-z 12614 df-uz 12879 df-xadd 13155 df-hash 14370 df-vtxdg 29484

This theorem is referenced by: vtxdgfusgrf 29515 eupth2lem3lem1 30247 eupth2lem3lem2 30248

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