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Theorem vtxd0nedgbfi 16105

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 5636	. . . 4 ⊢ (𝐷‘𝑈) = ((VtxDeg‘𝐺)‘𝑈)
3		vtxd0nedgb.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
4		vtxd0nedgb.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
5		eqid 2229	. . . . 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 16097	. . . 4 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) + (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}})))
11	2, 10	eqtrid 2274	. . 3 ⊢ (𝜑 → (𝐷‘𝑈) = ((♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) + (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}})))
12	11	eqeq1d 2238	. 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 16095	. . . . 5 ⊢ (𝜑 → {𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)} ∈ Fin)
14		hashcl 11033	. . . . 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 9446	. . 3 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}) ∈ ℝ)
17	15	nn0ge0d 9448	. . 3 ⊢ (𝜑 → 0 ≤ (♯‘{𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)}))
18	3, 4, 5, 6, 7, 8, 9	vtxlpfi 16096	. . . . 5 ⊢ (𝜑 → {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}} ∈ Fin)
19		hashcl 11033	. . . . 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 9446	. . 3 ⊢ (𝜑 → (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}) ∈ ℝ)
22	20	nn0ge0d 9448	. . 3 ⊢ (𝜑 → 0 ≤ (♯‘{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}}))
23		add20 8644	. . 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 1272	. 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 11045	. . . . . 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 11045	. . . . . 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 473	. . . 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 3522	. . . . . 6 ⊢ ({𝑖 ∈ dom 𝐼 ∣ 𝑈 ∈ (𝐼‘𝑖)} = ∅ ↔ ∀𝑖 ∈ dom 𝐼 ¬ 𝑈 ∈ (𝐼‘𝑖))
31		rabeq0 3522	. . . . . 6 ⊢ ({𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑈}} = ∅ ↔ ∀𝑖 ∈ dom 𝐼 ¬ (𝐼‘𝑖) = {𝑈})
32	30, 31	anbi12i 460	. . . . 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 757	. . . . . . 7 ⊢ (¬ (𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}) ↔ (¬ 𝑈 ∈ (𝐼‘𝑖) ∧ ¬ (𝐼‘𝑖) = {𝑈}))
35	34	ralbii 2536	. . . . . 6 ⊢ (∀𝑖 ∈ dom 𝐼 ¬ (𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}) ↔ ∀𝑖 ∈ dom 𝐼(¬ 𝑈 ∈ (𝐼‘𝑖) ∧ ¬ (𝐼‘𝑖) = {𝑈}))
36		ralnex 2518	. . . . . 6 ⊢ (∀𝑖 ∈ dom 𝐼 ¬ (𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}) ↔ ¬ ∃𝑖 ∈ dom 𝐼(𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}))
37		r19.26 2657	. . . . . 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 733	. . . . . . 7 ⊢ ((𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}) ↔ ((𝐼‘𝑖) = {𝑈} ∨ 𝑈 ∈ (𝐼‘𝑖)))
42		snidg 3696	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ {𝑈})
43		eleq2 2293	. . . . . . . . 9 ⊢ ((𝐼‘𝑖) = {𝑈} → (𝑈 ∈ (𝐼‘𝑖) ↔ 𝑈 ∈ {𝑈}))
44	42, 43	syl5ibrcom 157	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → ((𝐼‘𝑖) = {𝑈} → 𝑈 ∈ (𝐼‘𝑖)))
45		pm4.72 832	. . . . . . . 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 2531	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (∃𝑖 ∈ dom 𝐼(𝑈 ∈ (𝐼‘𝑖) ∨ (𝐼‘𝑖) = {𝑈}) ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))
49	48	notbid 671	. . . 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

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 713 = wceq 1395 ∈ wcel 2200 ∀wral 2508 ∃wrex 2509 {crab 2512 ∅c0 3492 {csn 3667 class class class wbr 4086 dom cdm 4723 ‘cfv 5324 (class class class)co 6013 Fincfn 6904 ℝcr 8021 0cc0 8022 + caddc 8025 ≤ cle 8205 ℕ₀cn0 9392 ♯chash 11027 Vtxcvtx 15853 iEdgciedg 15854 UPGraphcupgr 15932 VtxDegcvtxdg 16092

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-frec 6552 df-1o 6577 df-2o 6578 df-er 6697 df-en 6905 df-dom 6906 df-fin 6907 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-inn 9134 df-2 9192 df-3 9193 df-4 9194 df-5 9195 df-6 9196 df-7 9197 df-8 9198 df-9 9199 df-n0 9393 df-z 9470 df-dec 9602 df-uz 9746 df-xadd 9998 df-fz 10234 df-ihash 11028 df-ndx 13075 df-slot 13076 df-base 13078 df-edgf 15846 df-vtx 15855 df-iedg 15856 df-upgren 15934 df-vtxdg 16093

This theorem is referenced by: 1loopgrvd0fi 16112 1hevtxdg0fi 16113

W3C validator