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Theorem vtxd0nedgbfi 16706

Description: A vertex has degree 0 iff there is no edge incident with the vertex. (Contributed by AV, 24-Dec-2020.) (Revised by AV, 22-Mar-2021.)

**Hypotheses**
Ref	Expression
vtxd0nedgb.v	⊢ 𝑉 = (Vtx‘𝐺)
vtxd0nedgb.i	⊢ 𝐼 = (iEdg‘𝐺)
vtxd0nedgb.d	⊢ 𝐷 = (VtxDeg‘𝐺)
vtxd0nedgbfi.i	⊢ (𝜑 → dom 𝐼 ∈ Fin)
vtxd0nedgbfi.v	⊢ (𝜑 → 𝑉 ∈ Fin)
vtxd0nedgbfi.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
vtxd0nedgbfi.g	⊢ (𝜑 → 𝐺 ∈ UPGraph)

**Assertion**
Ref	Expression
vtxd0nedgbfi	⊢ (𝜑 → ((𝐷‘𝑈) = 0 ↔ ¬ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))

Distinct variable groups: 𝑖,𝐺 𝑖,𝐼 𝑈,𝑖 𝑖,𝑉 Allowed substitution hints: 𝜑(𝑖) 𝐷(𝑖)

**Proof of Theorem vtxd0nedgbfi**
Step	Hyp	Ref	Expression
1		vtxd0nedgb.d	. . . . 5 ⊢ 𝐷 = (VtxDeg‘𝐺)
2	1	fveq1i 5696	. . . 4 ⊢ (𝐷‘𝑈) = ((VtxDeg‘𝐺)‘𝑈)
3		vtxd0nedgb.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
4		vtxd0nedgb.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
5		eqid 2238	. . . . 5 ⊢ dom 𝐼 = dom 𝐼
6		vtxd0nedgbfi.i	. . . . 5 ⊢ (𝜑 → dom 𝐼 ∈ Fin)
7		vtxd0nedgbfi.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ Fin)
8		vtxd0nedgbfi.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
9		vtxd0nedgbfi.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
10	3, 4, 5, 6, 7, 8, 9	vtxdgfifival 16698	. . . 4 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) + (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}})))
11	2, 10	eqtrid 2283	. . 3 ⊢ (𝜑 → (𝐷‘𝑈) = ((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) + (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}})))
12	11	eqeq1d 2247	. 2 ⊢ (𝜑 → ((𝐷‘𝑈) = 0 ↔ ((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) + (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}})) = 0))
13	3, 4, 5, 6, 7, 8, 9	vtxedgfi 16696	. . . . 5 ⊢ (𝜑 → {𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)} ∈ Fin)
14		hashcl 11236	. . . . 5 ⊢ ({𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)} ∈ Fin → (♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) ∈ ℕ₀)
15	13, 14	syl 14	. . . 4 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) ∈ ℕ₀)
16	15	nn0red 9626	. . 3 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) ∈ ℝ)
17	15	nn0ge0d 9628	. . 3 ⊢ (𝜑 → 0 ≤ (♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}))
18	3, 4, 5, 6, 7, 8, 9	vtxlpfi 16697	. . . . 5 ⊢ (𝜑 → {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}} ∈ Fin)
19		hashcl 11236	. . . . 5 ⊢ ({𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}} ∈ Fin → (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}) ∈ ℕ₀)
20	18, 19	syl 14	. . . 4 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}) ∈ ℕ₀)
21	20	nn0red 9626	. . 3 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}) ∈ ℝ)
22	20	nn0ge0d 9628	. . 3 ⊢ (𝜑 → 0 ≤ (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}))
23		add20 8804	. . 3 ⊢ ((((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) ∈ ℝ ∧ 0 ≤ (♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)})) ∧ ((♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}) ∈ ℝ ∧ 0 ≤ (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}))) → (((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) + (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}})) = 0 ↔ ((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) = 0 ∧ (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}) = 0)))
24	16, 17, 21, 22, 23	syl22anc 1279	. 2 ⊢ (𝜑 → (((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) + (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}})) = 0 ↔ ((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) = 0 ∧ (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}) = 0)))
25		fihasheq0 11248	. . . . . 6 ⊢ ({𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)} ∈ Fin → ((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) = 0 ↔ {𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)} = ∅))
26	13, 25	syl 14	. . . . 5 ⊢ (𝜑 → ((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) = 0 ↔ {𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)} = ∅))
27		fihasheq0 11248	. . . . . 6 ⊢ ({𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}} ∈ Fin → ((♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}) = 0 ↔ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}} = ∅))
28	18, 27	syl 14	. . . . 5 ⊢ (𝜑 → ((♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}) = 0 ↔ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}} = ∅))
29	26, 28	anbi12d 477	. . . 4 ⊢ (𝜑 → (((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) = 0 ∧ (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}) = 0) ↔ ({𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)} = ∅ ∧ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}} = ∅)))
30		rabeq0 3552	. . . . . 6 ⊢ ({𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)} = ∅ ↔ ∀𝑖 ∈ dom 𝐼 ¬ 𝑈 ∈ (𝐼‘𝑖))
31		rabeq0 3552	. . . . . 6 ⊢ ({𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}} = ∅ ↔ ∀𝑖 ∈ dom 𝐼 ¬ (𝐼‘𝑖) = {𝑈})
32	30, 31	anbi12i 464	. . . . 5 ⊢ (({𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)} = ∅ ∧ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}} = ∅) ↔ (∀𝑖 ∈ dom 𝐼 ¬ 𝑈 ∈ (𝐼‘𝑖) ∧ ∀𝑖 ∈ dom 𝐼 ¬ (𝐼‘𝑖) = {𝑈}))
33	32	a1i 9	. . . 4 ⊢ (𝜑 → (({𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)} = ∅ ∧ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}} = ∅) ↔ (∀𝑖 ∈ dom 𝐼 ¬ 𝑈 ∈ (𝐼‘𝑖) ∧ ∀𝑖 ∈ dom 𝐼 ¬ (𝐼‘𝑖) = {𝑈})))
34		ioran 764	. . . . . . 7 ⊢ (¬ (𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}) ↔ (¬ 𝑈 ∈ (𝐼‘𝑖) ∧ ¬ (𝐼‘𝑖) = {𝑈}))
35	34	ralbii 2556	. . . . . 6 ⊢ (∀𝑖 ∈ dom 𝐼 ¬ (𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}) ↔ ∀𝑖 ∈ dom 𝐼(¬ 𝑈 ∈ (𝐼‘𝑖) ∧ ¬ (𝐼‘𝑖) = {𝑈}))
36		ralnex 2538	. . . . . 6 ⊢ (∀𝑖 ∈ dom 𝐼 ¬ (𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}) ↔ ¬ ∃𝑖 ∈ dom 𝐼(𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}))
37		r19.26 2677	. . . . . 6 ⊢ (∀𝑖 ∈ dom 𝐼(¬ 𝑈 ∈ (𝐼‘𝑖) ∧ ¬ (𝐼‘𝑖) = {𝑈}) ↔ (∀𝑖 ∈ dom 𝐼 ¬ 𝑈 ∈ (𝐼‘𝑖) ∧ ∀𝑖 ∈ dom 𝐼 ¬ (𝐼‘𝑖) = {𝑈}))
38	35, 36, 37	3bitr3ri 211	. . . . 5 ⊢ ((∀𝑖 ∈ dom 𝐼 ¬ 𝑈 ∈ (𝐼‘𝑖) ∧ ∀𝑖 ∈ dom 𝐼 ¬ (𝐼‘𝑖) = {𝑈}) ↔ ¬ ∃𝑖 ∈ dom 𝐼(𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}))
39	38	a1i 9	. . . 4 ⊢ (𝜑 → ((∀𝑖 ∈ dom 𝐼 ¬ 𝑈 ∈ (𝐼‘𝑖) ∧ ∀𝑖 ∈ dom 𝐼 ¬ (𝐼‘𝑖) = {𝑈}) ↔ ¬ ∃𝑖 ∈ dom 𝐼(𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈})))
40	29, 33, 39	3bitrd 214	. . 3 ⊢ (𝜑 → (((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) = 0 ∧ (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}) = 0) ↔ ¬ ∃𝑖 ∈ dom 𝐼(𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈})))
41		orcom 740	. . . . . . 7 ⊢ ((𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}) ↔ ((𝐼‘𝑖) = {𝑈} ∨ 𝑈 ∈ (𝐼‘𝑖)))
42		snidg 3738	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ {𝑈})
43		eleq2 2302	. . . . . . . . 9 ⊢ ((𝐼‘𝑖) = {𝑈} → (𝑈 ∈ (𝐼‘𝑖) ↔ 𝑈 ∈ {𝑈}))
44	42, 43	syl5ibrcom 157	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → ((𝐼‘𝑖) = {𝑈} → 𝑈 ∈ (𝐼‘𝑖)))
45		pm4.72 839	. . . . . . . 8 ⊢ (((𝐼‘𝑖) = {𝑈} → 𝑈 ∈ (𝐼‘𝑖)) ↔ (𝑈 ∈ (𝐼‘𝑖) ↔ ((𝐼‘𝑖) = {𝑈} ∨ 𝑈 ∈ (𝐼‘𝑖))))
46	44, 45	sylib 122	. . . . . . 7 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ (𝐼‘𝑖) ↔ ((𝐼‘𝑖) = {𝑈} ∨ 𝑈 ∈ (𝐼‘𝑖))))
47	41, 46	bitr4id 199	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → ((𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}) ↔ 𝑈 ∈ (𝐼‘𝑖)))
48	47	rexbidv 2551	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (∃𝑖 ∈ dom 𝐼(𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}) ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))
49	48	notbid 677	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (¬ ∃𝑖 ∈ dom 𝐼(𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}) ↔ ¬ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))
50	8, 49	syl 14	. . 3 ⊢ (𝜑 → (¬ ∃𝑖 ∈ dom 𝐼(𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}) ↔ ¬ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))
51	40, 50	bitrd 188	. 2 ⊢ (𝜑 → (((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) = 0 ∧ (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}) = 0) ↔ ¬ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))
52	12, 24, 51	3bitrd 214	1 ⊢ (𝜑 → ((𝐷‘𝑈) = 0 ↔ ¬ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 = wceq 1402 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 {crab 2532 ∅c0 3520 {csn 3709 class class class wbr 4130 dom cdm 4774 ‘cfv 5377 (class class class)co 6085 Fincfn 7022 ℝcr 8179 0cc0 8180 + caddc 8183 ≤ cle 8362 ℕ₀cn0 9568 ♯chash 11230 Vtxcvtx 16419 iEdgciedg 16420 UPGraphcupgr 16498 VtxDegcvtxdg 16693

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-1o 6687 df-2o 6688 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-5 9369 df-6 9370 df-7 9371 df-8 9372 df-9 9373 df-n0 9569 df-z 9650 df-dec 9783 df-uz 9932 df-xadd 10186 df-fz 10423 df-ihash 11231 df-ndx 13407 df-slot 13408 df-base 13410 df-edgf 16412 df-vtx 16421 df-iedg 16422 df-upgren 16500 df-vtxdg 16694

This theorem is used by: 1loopgrvd0fi 16713 1hevtxdg0fi 16714

W3C validator