Theorem clnbgrel 48326

Description: Characterization of a member 𝑁 of the closed neighborhood of a vertex 𝑋 in a graph 𝐺. (Contributed by AV, 9-May-2025.)

Hypotheses

Ref	Expression
clnbgrel.v	⊢ 𝑉 = (Vtx‘𝐺)
clnbgrel.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
clnbgrel	⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑋   𝑒,𝑉

Proof of Theorem clnbgrel

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clnbgrel.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	clnbgrcl 48319	. . 3 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) → 𝑋 ∈ 𝑉)
3	2	pm4.71ri 565	. 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋)))
4		clnbgrel.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐺)
5	1, 4	clnbgrval 48320	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑋) = ({𝑋} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
6	5	eleq2d 2826	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ 𝑁 ∈ ({𝑋} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒})))
7		elun 4090	. . . . 5 ⊢ (𝑁 ∈ ({𝑋} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}) ↔ (𝑁 ∈ {𝑋} ∨ 𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
8		elsn2g 4603	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ {𝑋} ↔ 𝑁 = 𝑋))
9		preq2 4673	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → {𝑋, 𝑛} = {𝑋, 𝑁})
10	9	sseq1d 3953	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ({𝑋, 𝑛} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ 𝑒))
11	10	rexbidv 3164	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
12	11	elrab 3636	. . . . . . 7 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
13	12	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
14	8, 13	orbi12d 924	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ((𝑁 ∈ {𝑋} ∨ 𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}) ↔ (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
15	7, 14	bitrid 284	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ ({𝑋} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}) ↔ (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
16		eleq1 2828	. . . . . . . . 9 ⊢ (𝑁 = 𝑋 → (𝑁 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
17	16	biimparc 480	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 𝑋) → 𝑁 ∈ 𝑉)
18		orc 873	. . . . . . . . 9 ⊢ (𝑁 = 𝑋 → (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
19	18	adantl 482	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 𝑋) → (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
20	17, 19	jca 516	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 𝑋) → (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
21	20	ex 413	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑁 = 𝑋 → (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
22		olc 874	. . . . . . . 8 ⊢ (∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒 → (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
23	22	anim2i 623	. . . . . . 7 ⊢ ((𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) → (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
24	23	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ((𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) → (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
25	21, 24	jaod 865	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ((𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)) → (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
26		orc 873	. . . . . . . 8 ⊢ (𝑁 = 𝑋 → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
27	26	a1i 11	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (𝑁 = 𝑋 → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
28		olc 874	. . . . . . . . 9 ⊢ ((𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
29	28	ex 413	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒 → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
30	29	adantl 482	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒 → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
31	27, 30	jaod 865	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → ((𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
32	31	expimpd 454	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ((𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)) → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
33	25, 32	impbid 213	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ((𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)) ↔ (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
34	6, 15, 33	3bitrd 306	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
35	34	pm5.32i 579	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋)) ↔ (𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
36		anass 469	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)) ↔ (𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
37	36	bicomi 225	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))) ↔ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
38		ancom 461	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ↔ (𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉))
39	37, 38	bianbi 633	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
40	3, 35, 39	3bitri 298	1 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  ∃wrex 3064  {crab 3392   ∪ cun 3888   ⊆ wss 3890  {csn 4562  {cpr 4564  ‘cfv 6492  (class class class)co 7363  Vtxcvtx 29090  Edgcedg 29141   ClNeighbVtx cclnbgr 48316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-clnbgr 48317

This theorem is referenced by:  clnbgrvtxel  48327  clnbgrisvtx  48328  clnbgrsym  48336  predgclnbgrel  48337  clnbgredg  48338  clnbgrgrimlem  48431  clnbgrgrim  48432