Theorem clnbgrel 47753

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 47746	. . 3 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) → 𝑋 ∈ 𝑉)
3	2	pm4.71ri 560	. 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋)))
4		clnbgrel.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐺)
5	1, 4	clnbgrval 47747	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑋) = ({𝑋} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
6	5	eleq2d 2825	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ 𝑁 ∈ ({𝑋} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒})))
7		elun 4163	. . . . 5 ⊢ (𝑁 ∈ ({𝑋} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}) ↔ (𝑁 ∈ {𝑋} ∨ 𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
8		elsn2g 4669	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ {𝑋} ↔ 𝑁 = 𝑋))
9		preq2 4739	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → {𝑋, 𝑛} = {𝑋, 𝑁})
10	9	sseq1d 4027	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ({𝑋, 𝑛} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ 𝑒))
11	10	rexbidv 3177	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
12	11	elrab 3695	. . . . . . 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 918	. . . . 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 283	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ ({𝑋} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}) ↔ (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
16		eleq1 2827	. . . . . . . . 9 ⊢ (𝑁 = 𝑋 → (𝑁 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
17	16	biimparc 479	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 𝑋) → 𝑁 ∈ 𝑉)
18		orc 867	. . . . . . . . 9 ⊢ (𝑁 = 𝑋 → (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
19	18	adantl 481	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 𝑋) → (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
20	17, 19	jca 511	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 𝑋) → (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
21	20	ex 412	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑁 = 𝑋 → (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
22		olc 868	. . . . . . . 8 ⊢ (∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒 → (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
23	22	anim2i 617	. . . . . . 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 859	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ((𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)) → (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
26		orc 867	. . . . . . . 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 868	. . . . . . . . 9 ⊢ ((𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
29	28	ex 412	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒 → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
30	29	adantl 481	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒 → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
31	27, 30	jaod 859	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → ((𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
32	31	expimpd 453	. . . . 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 212	. . . 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 305	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
35	34	pm5.32i 574	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋)) ↔ (𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
36		anass 468	. . . 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 224	. . 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 460	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ↔ (𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉))
39	37, 38	bianbi 627	. 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 297	1 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  {crab 3433   ∪ cun 3961   ⊆ wss 3963  {csn 4631  {cpr 4633  ‘cfv 6563  (class class class)co 7431  Vtxcvtx 29028  Edgcedg 29079   ClNeighbVtx cclnbgr 47743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-clnbgr 47744

This theorem is referenced by:  clnbgrvtxel  47754  clnbgrisvtx  47755  clnbgrsym  47762  predgclnbgrel  47763  clnbgredg  47764  clnbgrgrimlem  47839  clnbgrgrim  47840